Another example graphing with derivatives

WEBVTT
Kind: captions
Language: en

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Let's say we've got the
function, f of x is equal

00:00:04.290 --> 00:00:08.780
to the natural log of x
to the fourth plus 27.

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And all we want to do is take
its first and second

00:00:11.060 --> 00:00:15.200
derivatives, and use as much of
our techniques as we have at

00:00:15.200 --> 00:00:17.320
our disposal to attempt
to graph it without a

00:00:17.320 --> 00:00:18.210
graphing calculator.

00:00:18.210 --> 00:00:20.560
If we have time, I'll take out
the graphing calculator and

00:00:20.560 --> 00:00:22.810
see if our answer matches up.

00:00:22.810 --> 00:00:25.300
So a good place to start
is to take the first

00:00:25.300 --> 00:00:26.420
derivative of this.

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So let me do that over here.

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So the derivative of f.

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Well, you take the derivative
of the inside, so take the

00:00:34.870 --> 00:00:39.900
derivative of that right there,
which is 4x to the third, and

00:00:39.900 --> 00:00:44.960
then multiply it times the
derivative of the outside,

00:00:44.960 --> 00:00:46.400
with respect to the inside.

00:00:46.400 --> 00:00:48.940
So the derivative of the
natural log of x is 1over x.

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So the derivative of this whole
thing with respect to this

00:00:51.580 --> 00:00:56.040
inside expression is going
to be, so times 1 over x

00:00:56.040 --> 00:00:58.220
to the fourth plus 27.

00:00:58.220 --> 00:01:01.660
If you found that confusing,
you might want to rewatch

00:01:01.660 --> 00:01:03.460
the chain rule videos.

00:01:03.460 --> 00:01:05.290
But that's the first
derivative of our function.

00:01:05.290 --> 00:01:10.670
I could rewrite this, this is
equal to 4x to the third over

00:01:10.670 --> 00:01:12.850
x to the fourth, plus 27.

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Or I could write it as 4x to
the third times x to the

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fourth, plus 27 to
the negative 1.

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All three of these
expressions are equivalent.

00:01:23.130 --> 00:01:25.950
I'm just the writing, I
multiplied it out, or I could

00:01:25.950 --> 00:01:27.670
write this as a negative
exponent, or I could write

00:01:27.670 --> 00:01:29.870
this as a fraction, with
this in the denominator.

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They're all the equivalent.

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So that's our first derivative.

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Let's do our second derivative.

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Our second derivative,
this looks like it'll get

00:01:36.990 --> 00:01:39.470
a little bit hairier.

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So our second derivative is
the derivative of this.

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So it's equal to, we can now
use the product rule, is the

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derivative of this first
expression, times the

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second expression.

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So the derivative of this first
expression, 3 times 4is 12.

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12x squared, right, we just
decrement the 3 by 1, times the

00:01:58.450 --> 00:02:06.280
second expression, times x to
the fourth plus 27 to the minus

00:02:06.280 --> 00:02:12.380
1, and then to that, we want to
add just the first expression,

00:02:12.380 --> 00:02:15.900
not its derivative, so just 4x
to the third, times

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the derivative of the
second expression.

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And the derivative of the
second expression, we could

00:02:19.380 --> 00:02:22.940
take the derivative of the
inside, which is just 4x to the

00:02:22.940 --> 00:02:28.690
third, the derivative of 27 is
just 0, so times 4x to the

00:02:28.690 --> 00:02:32.670
third, times the derivative
of this whole thing with

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respect to the inside.

00:02:33.790 --> 00:02:37.650
So times, so you take this
exponent, put out front, so

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times minus 1, times this whole
thing, x to the fourth plus

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27 to the, we decrement this
by one more, so minus two.

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So let's see if I can simplify
this expression a little bit.

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So this is equal to, so this
right here is equal to 12x

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squared over this thing, x to
the fourth plus 27, and then,

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let's see, if we multiply we're
going to have a minus here, so

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it's a minus, you multiply
these two guys, 4 times 4is 16,

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16x to the third times x to the
third is x to the sixth,

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over this thing squared.

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Over x to the fourth
plus 27 squared.

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That's just another way to
rewrite that expression

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right there, right?

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To the minus 2, you just put in
the denominator and make it

00:03:31.810 --> 00:03:34.150
into a positive 2 in
the denominator.

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Same thing.

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Now, if you've seen these
problems in the past, we

00:03:37.860 --> 00:03:39.480
always want to set these
things equal to 0.

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We want to solve
for x equals 0.

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So it'll be useful to have this
expressed as just one fraction,

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instead of the difference or
the sum of two fractions.

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So what we can do, is we could
have a common denominator.

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So we could multiply both the
numerator and denominator of

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this expression by x to the
fourth plus 27, and

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what do we get?

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So this is equal to, so if we
multiply this first expression,

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times x to the fourth plus 27,
we get 12x squared, times

00:04:06.730 --> 00:04:09.290
x to the fourth, plus 27.

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And then in the denominator,
you have x to the

00:04:12.340 --> 00:04:15.830
fourth plus 27 squared.

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All I did, I multiplied this
numerator and this denominator

00:04:18.220 --> 00:04:19.610
by x to the the fourth plus 27.

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I didn't change it.

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And then we have
that second term.

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Minus 16 x to the sixth over x
to the fourth plus 27 squared.

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The whole reason why did that?

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Now I have a common
denominator, now I can

00:04:32.740 --> 00:04:34.250
just add the numerators.

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So this is going to be
equal to, let's see.

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The denominator, we know what
the denominator is, it is x to

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the fourth plus 27 squared.

00:04:48.410 --> 00:04:49.330
That's our denominator.

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And then we can
multiply this out.

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This is 12x squared
times x to the fourth.

00:04:52.580 --> 00:04:59.040
That's 12x to the sixth,
plus 27 times 12.

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I don't even feel like
multiplying 27 times 12,so

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I'll just write that out.

00:05:02.060 --> 00:05:09.300
So plus 27 times 12x squared, I
just multiplied the 12x squared

00:05:09.300 --> 00:05:14.570
times the 27, and then minus
16x to the sixth minus

00:05:14.570 --> 00:05:17.430
16x to the sixth.

00:05:17.430 --> 00:05:20.000
And this simplifies to,
let's see if I can simplify

00:05:20.000 --> 00:05:21.240
this even further.

00:05:21.240 --> 00:05:24.050
7x to the sixth here,
x to the sixth here.

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So this is equal to,
do this in pink.

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This is equal to the 27 times
12x squared, I don't feel like

00:05:31.220 --> 00:05:35.970
figuring that out right now,
times 12x squared, and then you

00:05:35.970 --> 00:05:39.140
have minus 16x to the sixth
and plus 12x to the sixth.

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So you add those two,
you get minus 4.

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12 minus 15is minus 4, x to the
sixth, all of that over x to

00:05:48.900 --> 00:05:57.160
the fourth plus 27
plus 27 squared.

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And that is our
second derivative.

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Now, we've done all of the
derivatives, and this was

00:06:02.190 --> 00:06:03.800
actually a pretty
hairy problem.

00:06:03.800 --> 00:06:06.860
And now we can solve for when
the first and the second

00:06:06.860 --> 00:06:10.305
derivatives equal 0, and we'll
have our candidate, well, we'll

00:06:10.305 --> 00:06:12.810
know our critical points, and
then we'll have our candidate

00:06:12.810 --> 00:06:14.780
inflection points, and
see if we can make any

00:06:14.780 --> 00:06:15.630
headway from there.

00:06:15.630 --> 00:06:18.940
So first, let's see where our
first derivative is equal to 0,

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and get our critical points.

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Or at least maybe, also
maybe, where it's undefined.

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So this is equal to 0.

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If we want to set, if the only
place that this can equal to 0

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is if this numerator
is equal to 0.

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This denominator, actually, if
we are assuming we're dealing

00:06:34.490 --> 00:06:37.490
with real numbers, this term
right here is always going to

00:06:37.490 --> 00:06:41.130
be greater than or equal to 0
for any value of x, because

00:06:41.130 --> 00:06:42.490
it's an even exponent.

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So this thing can never equals
0, right, because you're

00:06:44.790 --> 00:06:47.410
adding 27 to something
that's non-negative.

00:06:47.410 --> 00:06:49.540
So this will never equal
0, so this will also

00:06:49.540 --> 00:06:50.850
never be undefined.

00:06:50.850 --> 00:06:54.890
So there's no undefined
critical points here, but

00:06:54.890 --> 00:06:58.550
we could set the numerator
equal to 0 pretty easily.

00:06:58.550 --> 00:07:02.060
If we wanted to set this equal
to 0, we just say 4x to the

00:07:02.060 --> 00:07:04.900
third is equal to 0, and we
know what x-value will make

00:07:04.900 --> 00:07:08.280
that equal to 0, x has
to be equal to 0.

00:07:08.280 --> 00:07:10.230
4 times something to the
third is equal to 0, that

00:07:10.230 --> 00:07:11.760
something has to be 0.

00:07:11.760 --> 00:07:14.770
x to the third has to
be 0, x has to be 0.

00:07:14.770 --> 00:07:19.590
So we can write, f prime
of 0 is equal to 0.

00:07:19.590 --> 00:07:22.350
So 0 is a critical point.

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0 is a critical point.

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The slope at 0 is 0.

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We don't know if it's a
maximum or a minimum, or

00:07:32.060 --> 00:07:33.380
an inflection point yet.

00:07:36.530 --> 00:07:37.785
We'll explore it a
little bit more.

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And actually, just so
we get the coordinate,

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what's the coordinate?

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The coordinate x is 0, and then
y is the natural log-- if x is

00:07:45.150 --> 00:07:49.620
0, this just turns out, it's
the a natural log of 27.

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Let me figure out what that is,
I'll get the calculator out.

00:07:51.880 --> 00:07:53.740
I said I wouldn't use a
graphing calculator, but I can

00:07:53.740 --> 00:07:54.910
use a regular calculator.

00:07:54.910 --> 00:07:59.870
So27, if I were to take the
natural log of that, for

00:07:59.870 --> 00:08:01.420
our purposes let's
just call it 3.3.

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We're just trying to get the
general shape of the graph.

00:08:03.760 --> 00:08:04.190
So 3.3.

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Well, we could just say
2.9 and it kept going.

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So this is a critical
point right here.

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The slope is 0 here.

00:08:15.210 --> 00:08:20.260
Slope is equal to 0
at x is equal to 0.

00:08:20.260 --> 00:08:22.110
So this is one thing
we want to block off.

00:08:22.110 --> 00:08:23.930
And let's see if we can
find any candidate

00:08:23.930 --> 00:08:25.170
inflection points.

00:08:25.170 --> 00:08:27.910
And remember, candidate
inflection points are where the

00:08:27.910 --> 00:08:29.510
second derivative equals 0.

00:08:29.510 --> 00:08:32.040
Now if the second derivative
equals 0, that doesn't tell

00:08:32.040 --> 00:08:33.790
us that those are definitely
inflection points.

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Let me make this very clear.

00:08:35.910 --> 00:08:39.520
If, let me do it
in a new color.

00:08:39.520 --> 00:08:52.160
If x is inflection, then the
second derivative at x is

00:08:52.160 --> 00:08:54.090
going to be equal to 0.

00:08:54.090 --> 00:08:58.910
Because you're having
a change concavity.

00:08:58.910 --> 00:09:02.830
You have a change in the slope,
goes from either increasing

00:09:02.830 --> 00:09:05.340
to decreasing or from
decreasing to increasing.

00:09:05.340 --> 00:09:09.350
But if the derivative is equal
to 0, the second derivative is

00:09:09.350 --> 00:09:11.900
equal to 0, you cannot assume
that is an inflection point.

00:09:11.900 --> 00:09:14.390
So what we're going to do is,
we're going to find all of the

00:09:14.390 --> 00:09:19.140
point at which this is true,
and then see if we actually do

00:09:19.140 --> 00:09:22.230
have a sign change in the
second derivative of that

00:09:22.230 --> 00:09:25.280
point, and only if you have a
sign change, then you can say

00:09:25.280 --> 00:09:26.550
it's an inflection point.

00:09:26.550 --> 00:09:28.170
So let's see if we can do that.

00:09:28.170 --> 00:09:30.220
So just because a second
derivative is 0, that by

00:09:30.220 --> 00:09:32.550
itself does not tell you
it's an inflection point.

00:09:32.550 --> 00:09:36.210
It has to have a second
derivative of 0, and when you

00:09:36.210 --> 00:09:40.190
go above or below that x, the
second derivative has to

00:09:40.190 --> 00:09:41.500
actually change signs.

00:09:41.500 --> 00:09:42.140
Only then.

00:09:42.140 --> 00:09:54.170
So we can say, if f prime
changes signs around x,

00:09:54.170 --> 00:09:56.890
then we can say that
x is an inflection.

00:10:00.226 --> 00:10:03.070
And if it's changing signs
around x, then it's definitely

00:10:03.070 --> 00:10:06.470
going to be 0 right at x, but
you have to actually see that

00:10:06.470 --> 00:10:09.350
if it's negative before x, has
to be positive after x,or if

00:10:09.350 --> 00:10:11.737
it's positive before x, has
to be negative after x.

00:10:11.737 --> 00:10:13.270
So let's test that out.

00:10:13.270 --> 00:10:15.610
So the first thing we
need to do is find

00:10:15.610 --> 00:10:16.490
these candidate points.

00:10:16.490 --> 00:10:19.430
Remember, the candidate
points are where the second

00:10:19.430 --> 00:10:21.160
derivative is equal to 0.

00:10:21.160 --> 00:10:23.570
We're going to find those
points, and then see if this

00:10:23.570 --> 00:10:25.670
is true, that the sign
actually changes.

00:10:25.670 --> 00:10:29.750
We want to find where this
thing over here is equal to 0.

00:10:29.750 --> 00:10:32.450
And once again, for this to
be equal to 0, the numerator

00:10:32.450 --> 00:10:33.390
has to be equal to 0.

00:10:33.390 --> 00:10:36.200
This denominator can never be
equal to 0 if we're dealing

00:10:36.200 --> 00:10:38.570
with real numbers, which I
think is a fair assumption.

00:10:38.570 --> 00:10:43.450
So let's see where this our
numerator can be equal to 0

00:10:43.450 --> 00:10:45.650
for the second derivative.

00:10:45.650 --> 00:10:48.610
So let's set the numerator
of the second derivative.

00:10:48.610 --> 00:10:54.900
27 times 12x squared minus 4x
to the sixth is equal to 0.

00:10:54.900 --> 00:10:56.210
Remember, that's just
the numerator of our

00:10:56.210 --> 00:10:57.250
second derivative.

00:10:57.250 --> 00:10:59.360
Any x that makes the
numerator 0 is making

00:10:59.360 --> 00:11:01.580
the second derivative 0.

00:11:01.580 --> 00:11:06.500
So let's factor
out a 4x squared.

00:11:06.500 --> 00:11:09.060
So 4x squared.

00:11:09.060 --> 00:11:14.140
Now we'll have 27 times, if we
factor 4 out of the 12, we'll

00:11:14.140 --> 00:11:17.900
just get a 3, and we factored
out the x squared, minus, we

00:11:17.900 --> 00:11:21.120
factored out the 4, we factored
out an x squared, so we have x

00:11:21.120 --> 00:11:24.580
to the fourth is equal to 0.

00:11:24.580 --> 00:11:29.480
So the x's that will make this
equal to 0 will satisfy either,

00:11:29.480 --> 00:11:35.180
I'll switch colors, either 4x
squared is equal to 0, or, now

00:11:35.180 --> 00:11:37.060
27 times 3, I can do
that in my head.

00:11:37.060 --> 00:11:38.170
That's 81.

00:11:38.170 --> 00:11:43.060
20 times 3 is 60, 7 times
3 is 21, 60 plus 21is 81.

00:11:43.060 --> 00:11:48.920
Or 81 minus x to the
fourth is equal to 0.

00:11:48.920 --> 00:11:51.410
Any x that satisfies either
of these will make this

00:11:51.410 --> 00:11:52.700
entire expression equal 0.

00:11:52.700 --> 00:11:54.180
Because if this thing is
0, the whole thing is

00:11:54.180 --> 00:11:54.755
going to be equal to 0.

00:11:54.755 --> 00:11:56.750
If this thing is 0, the
whole thing is going

00:11:56.750 --> 00:11:57.445
to be equal to 0.

00:11:57.445 --> 00:11:59.640
Let me be clear, this
is 81 right there.

00:11:59.640 --> 00:12:00.760
So let's solve this.

00:12:00.760 --> 00:12:07.460
This is going to be 0 when
x is equal to 0, itself.

00:12:07.460 --> 00:12:10.560
This is going to be equal
to 0 when x, let's see.

00:12:10.560 --> 00:12:12.590
If we add x to the fourth to
both sides, you get x to

00:12:12.590 --> 00:12:15.060
the fourth is equal to 81.

00:12:15.060 --> 00:12:18.070
If we take the square root of
both sides of this, you get x

00:12:18.070 --> 00:12:23.090
squared is equal to 9, or so
you get x is plus or minus 3.

00:12:23.090 --> 00:12:29.590
x is equal to plus
or minus three.

00:12:29.590 --> 00:12:32.640
So these are our candidate
inflection points, x is equal

00:12:32.640 --> 00:12:36.770
to 0, x is equal to plus 3,
or x is equal to minus 3.

00:12:36.770 --> 00:12:39.150
So what we have to do now, is
to see whether the second

00:12:39.150 --> 00:12:42.360
derivative changes signs around
these points in order to be

00:12:42.360 --> 00:12:44.710
able to label them
inflection points.

00:12:44.710 --> 00:12:49.130
So what happens when x
is slightly below 0?

00:12:49.130 --> 00:12:52.290
So let's take the situation,
let's do all the scenarios.

00:12:52.290 --> 00:12:54.720
What happens when x
is slightly below 0?

00:12:54.720 --> 00:12:58.620
Not all of them, necessarily,
but if x is like 0.1.

00:12:58.620 --> 00:13:01.490
What is the second derivative
going to be doing?

00:13:01.490 --> 00:13:07.230
If x is 0.1, or if x is minus
0.1, this term right here is

00:13:07.230 --> 00:13:12.080
going to be positive, and
then this is going to be 81

00:13:12.080 --> 00:13:13.990
minus 0.1 to the fourth.

00:13:13.990 --> 00:13:16.890
So that's going to be a
very small number, right?

00:13:16.890 --> 00:13:19.520
So it's going to be some
positive number times 81

00:13:19.520 --> 00:13:20.530
minus a small number.

00:13:20.530 --> 00:13:22.300
So it's going to be
a positive number.

00:13:22.300 --> 00:13:26.930
So when x is less than 0, or
just slightly less than 0, our

00:13:26.930 --> 00:13:31.740
second derivative is positive.

00:13:31.740 --> 00:13:34.520
Now what happens when
x is slightly larger?

00:13:34.520 --> 00:13:36.320
When I write this notation, I
want to be careful, I mean,

00:13:36.320 --> 00:13:39.960
really, just right below 0.

00:13:39.960 --> 00:13:43.030
Now when x is right
above 0, what happens?

00:13:43.030 --> 00:13:47.040
Let's say x was 0.01,
or 0.1, positive 0.1.

00:13:47.040 --> 00:13:48.180
Well, it's going to
be the same thing.

00:13:48.180 --> 00:13:49.830
Because in both cases,
we're squaring, and

00:13:49.830 --> 00:13:50.760
we're taking the fourth.

00:13:50.760 --> 00:13:53.850
So you're kind of losing
your sign information.

00:13:53.850 --> 00:13:57.290
So if x is 0.1, this thing
is going to be a small

00:13:57.290 --> 00:13:58.470
positive number.

00:13:58.470 --> 00:14:00.850
You're going to be subtracting
a very small positive number

00:14:00.850 --> 00:14:05.060
from 81, but 81 minus a small
number is still going

00:14:05.060 --> 00:14:05.920
to be positive.

00:14:05.920 --> 00:14:09.810
So you're going to positive
times a positive, so your

00:14:09.810 --> 00:14:13.530
second derivative is still
going to be greater than 0.

00:14:13.530 --> 00:14:15.430
So something interesting here.

00:14:15.430 --> 00:14:20.940
f at your second derivative is
0 when x is equal to 0, but it

00:14:20.940 --> 00:14:22.980
is a not an inflection point.

00:14:22.980 --> 00:14:29.230
Because notice, the concavity
did not change around 0.

00:14:29.230 --> 00:14:32.810
Our second derivative is
positive as we approach 0 from

00:14:32.810 --> 00:14:36.290
the left, and it's positive as
we approach 0 from the right.

00:14:36.290 --> 00:14:41.340
So in general, at 0, we're
always, as we're near 0 from

00:14:41.340 --> 00:14:44.040
either direction, we're going
to be concave upwards.

00:14:44.040 --> 00:14:48.820
So the fact that 0 is a
critical point, and that we're

00:14:48.820 --> 00:14:52.600
always concave upward, as we
approach 0 from either side,

00:14:52.600 --> 00:14:55.690
this tells us that this
is a minimum point.

00:15:00.760 --> 00:15:04.600
Because we're concave
upwards all around 0.

00:15:04.600 --> 00:15:07.160
So 0 is not an
inflection point.

00:15:07.160 --> 00:15:09.490
Let's see if positive
and negative 3 are

00:15:09.490 --> 00:15:11.270
inflection points.

00:15:11.270 --> 00:15:14.520
And if you study this
equation, let me write

00:15:14.520 --> 00:15:17.480
our-- and actually, I
just want to be clear.

00:15:17.480 --> 00:15:19.050
I've just been using
the numerator of the

00:15:19.050 --> 00:15:20.270
second derivative.

00:15:20.270 --> 00:15:23.880
The whole second derivative is
this thing right here, but I've

00:15:23.880 --> 00:15:25.800
been ignoring the denominator
because the denominator

00:15:25.800 --> 00:15:27.340
is always positive.

00:15:27.340 --> 00:15:29.140
So if we're trying to
understand whether things are

00:15:29.140 --> 00:15:30.980
positive or negative, we just
really have to determine

00:15:30.980 --> 00:15:33.000
whether the numerator is
positive or negative.

00:15:33.000 --> 00:15:35.430
Because this expression right
there is always positive.

00:15:35.430 --> 00:15:38.930
It's something to
the second power.

00:15:38.930 --> 00:15:43.200
So let's test whether we have a
change in concavity around x is

00:15:43.200 --> 00:15:45.610
equal to positive
or negative 3.

00:15:45.610 --> 00:15:48.450
So remember, the numerator of
our, let me just rewrite our

00:15:48.450 --> 00:15:50.850
second derivative, just so
you see it here. f

00:15:50.850 --> 00:15:52.180
prime prime of x.

00:15:52.180 --> 00:15:54.170
The numerator is this
thing right here.

00:15:54.170 --> 00:16:00.330
It's 4x squared times 81
minus x to the fourth.

00:16:00.330 --> 00:16:03.050
and the denominator
was up here, x to the

00:16:03.050 --> 00:16:04.430
fourth plus 27 squared.

00:16:10.540 --> 00:16:12.120
That was our second derivative.

00:16:12.120 --> 00:16:16.270
Let's see if this changes signs
around positive or negative 3.

00:16:16.270 --> 00:16:19.940
And actually, we should get the
same answer, because regardless

00:16:19.940 --> 00:16:22.440
of whether we put positive
or negative 3 here.

00:16:22.440 --> 00:16:24.360
you lose all your sign
information because you're

00:16:24.360 --> 00:16:25.835
taking it to the fourth
power, you're taking it

00:16:25.835 --> 00:16:26.630
to the second power.

00:16:26.630 --> 00:16:28.850
And obviously, anything to the
fourth power is always going to

00:16:28.850 --> 00:16:30.990
be positive, anything to the
second power is always

00:16:30.990 --> 00:16:31.710
going to be negative.

00:16:31.710 --> 00:16:35.170
So when we do our test, if it's
true for positive 3, it's

00:16:35.170 --> 00:16:37.950
probably going to be true
for negative 3 as well.

00:16:37.950 --> 00:16:39.540
But let's just try it out.

00:16:39.540 --> 00:16:44.960
So when x is just a little bit
less than positive 3, what's

00:16:44.960 --> 00:16:46.820
the sign of f prime prime of x?

00:16:46.820 --> 00:16:51.970
So it's going to be 4 times
9, or it's going to be 4

00:16:51.970 --> 00:16:53.880
times a positive number.

00:16:53.880 --> 00:16:55.930
It might be like 2.999,
but this is still

00:16:55.930 --> 00:16:57.420
going to be positive.

00:16:57.420 --> 00:17:01.540
So this is going to be positive
when x is approaching 3, and

00:17:01.540 --> 00:17:05.360
then this is going to be, well,
if x is 3, this is 0, so x is

00:17:05.360 --> 00:17:07.420
a little bit less than 3.

00:17:07.420 --> 00:17:11.390
If x is a little bit less than
3, if it's like 2.9999, this

00:17:11.390 --> 00:17:14.120
number is going to be less
than 81, so this is also

00:17:14.120 --> 00:17:15.460
going to be positive.

00:17:15.460 --> 00:17:17.860
And of course, the denominator
is always positive.

00:17:17.860 --> 00:17:21.490
So as x is less than 3, is
approaching from the left,

00:17:21.490 --> 00:17:22.760
we are concave upwards.

00:17:22.760 --> 00:17:24.650
This thing's going
to be a positive.

00:17:24.650 --> 00:17:27.610
Then f prime prime
is greater than 0.

00:17:27.610 --> 00:17:31.380
We are upwards,
concave upwards.

00:17:31.380 --> 00:17:35.230
When x is just larger than
3, what's going to happen?

00:17:35.230 --> 00:17:37.820
Well, this first term is
still going to be positive.

00:17:37.820 --> 00:17:41.220
But if x is just larger than 3,
x to the fourth is going to be

00:17:41.220 --> 00:17:44.720
just larger than 81, and so
this second term is going to be

00:17:44.720 --> 00:17:46.130
negative in that situation.

00:17:48.676 --> 00:17:51.150
Let me do it ina new color.

00:17:51.150 --> 00:17:53.910
It's going to be negative
when x is larger than 3.

00:17:53.910 --> 00:17:56.360
Because this is going
to be larger than 81.

00:17:56.360 --> 00:17:59.230
So if this is negative and this
is positive, then the whole

00:17:59.230 --> 00:18:00.980
thing is going to be negative,
because this denominator is

00:18:00.980 --> 00:18:02.440
still going to be positive.

00:18:02.440 --> 00:18:05.251
So then f prime prime is going
to be less than 0, so we're

00:18:05.251 --> 00:18:09.290
going to be concave downwards.

00:18:09.290 --> 00:18:10.700
One last one.

00:18:10.700 --> 00:18:16.280
What happens when x is just
a greater than minus 3?

00:18:16.280 --> 00:18:19.470
So just being greater
than minus 3, that's

00:18:19.470 --> 00:18:22.940
like minus 2.99999.

00:18:22.940 --> 00:18:25.280
So when you take minus 2.99
square it, you're going to get

00:18:25.280 --> 00:18:27.760
a positive number, so this
is going to be positive.

00:18:27.760 --> 00:18:31.010
And if you take minus 2.99 to
the fourth, that's going to be

00:18:31.010 --> 00:18:32.640
a little bit less
than 81, right?

00:18:32.640 --> 00:18:35.130
Because 2.99 to the fourth is a
little bit less than 81, so

00:18:35.130 --> 00:18:37.030
this is still going
to be positive.

00:18:37.030 --> 00:18:39.920
So you have a positive times a
positive divided by a positive,

00:18:39.920 --> 00:18:43.120
so you're going to be concave
upwards, because your second

00:18:43.120 --> 00:18:45.090
derivative is going to
be greater than 0.

00:18:45.090 --> 00:18:47.740
Concave upwards.

00:18:47.740 --> 00:18:52.000
And then finally, when x is
just, just less than negative

00:18:52.000 --> 00:18:55.100
3, remember, when I write this
down, I don't mean for all x's

00:18:55.100 --> 00:18:58.770
larger than negative 3,or all
x's smaller than negative 3.

00:18:58.770 --> 00:19:01.130
There's actually no, well, I
can't think of the notation

00:19:01.130 --> 00:19:04.910
that would say just, as we just
approach three in this case,

00:19:04.910 --> 00:19:09.120
from the left, but what happens
if we just go to minus 3.11?

00:19:09.120 --> 00:19:12.570
Or 3.01, I guess is a
better one, or 3.1?

00:19:12.570 --> 00:19:15.040
Well, this term right here
is going to be positive.

00:19:15.040 --> 00:19:19.320
But if we take minus 3.1 to the
fourth, that's going to be

00:19:19.320 --> 00:19:21.760
larger than positive 81, right?

00:19:21.760 --> 00:19:24.080
The sign will become positive,
it'll be larger than 81, so

00:19:24.080 --> 00:19:26.210
this'll become negative.

00:19:26.210 --> 00:19:28.540
So in that case as well, we'll
have a positive times a

00:19:28.540 --> 00:19:32.410
negative divided by a positive,
so then our second derivative

00:19:32.410 --> 00:19:34.100
is going to be negative.

00:19:34.100 --> 00:19:35.830
And so we're going
to be downwards.

00:19:38.340 --> 00:19:42.060
So I think we're ready to plot.
so first of all, is x plus or

00:19:42.060 --> 00:19:43.120
minus 3 inflection points?

00:19:43.120 --> 00:19:43.870
Sure!

00:19:43.870 --> 00:19:47.800
As we approach x is equal to 3
from the left, we are concave

00:19:47.800 --> 00:19:52.810
upwards, and then as we cross
3, the second derivative is 0.

00:19:52.810 --> 00:19:54.550
The second derivative's
0, I lost it up here.

00:19:54.550 --> 00:19:57.880
The second derivative is 0.

00:19:57.880 --> 00:20:00.590
And then, as we go to the
right of 3, we become

00:20:00.590 --> 00:20:01.400
concave downwards.

00:20:01.400 --> 00:20:04.050
So we got our sign change
in the second derivative.

00:20:04.050 --> 00:20:06.000
So x is equal to 3.

00:20:06.000 --> 00:20:08.610
So 3 is definitely an
inflection point, and the

00:20:08.610 --> 00:20:10.390
same argument could be
made for negative 3.

00:20:10.390 --> 00:20:12.770
We switch signs as we cross 3.

00:20:12.770 --> 00:20:17.620
So these definitely are
inflection points.

00:20:17.620 --> 00:20:19.920
Just so we get the exact
coordinates, let's figure out

00:20:19.920 --> 00:20:24.050
what f of 3 is, or f of
positive and negative 3.

00:20:24.050 --> 00:20:25.840
And then we're ready to graph.

00:20:25.840 --> 00:20:30.110
So first of all, we know that
f, we know that the point 0,

00:20:30.110 --> 00:20:34.880
3.29, that this was a minimum.

00:20:34.880 --> 00:20:37.870
Because 0 was a critical
point, the slope is 0 there,

00:20:37.870 --> 00:20:41.000
and because it's concave
upwards all around 0.

00:20:41.000 --> 00:20:43.230
So 0 is definitely not
an inflection point.

00:20:43.230 --> 00:20:46.970
And then we know that the
points positive 3 and minus 3

00:20:46.970 --> 00:20:50.210
are inflection points, and in
order to figure out their

00:20:50.210 --> 00:20:53.110
y-coordinates, we can
just evaluate them.

00:20:53.110 --> 00:20:54.950
So they're actually going to
have the same y-coordinates,

00:20:54.950 --> 00:20:57.680
because if you put a minus3 or
positive 3 and take it to the

00:20:57.680 --> 00:20:59.010
fourth power, you're going
to get the same thing.

00:20:59.010 --> 00:21:01.090
Let's figure out what they are.

00:21:01.090 --> 00:21:04.770
So if we take 3 to the fourth
power, that's what, 81.

00:21:04.770 --> 00:21:09.990
81 plus 27 is equal to 108,
and then we want to take

00:21:09.990 --> 00:21:11.160
the natural log of it.

00:21:14.730 --> 00:21:17.130
Let's just say 4.7, just
to get a rough idea.

00:21:17.130 --> 00:21:18.780
That's4.7.

00:21:18.780 --> 00:21:20.920
And that's true of whether we
do positive or negative 3,

00:21:20.920 --> 00:21:22.300
because we took to
the fourth power.

00:21:22.300 --> 00:21:27.720
So it's 4.7, 4.7.

00:21:27.720 --> 00:21:31.240
These are both
inflection points.

00:21:31.240 --> 00:21:33.570
And we should be
ready to graph it!

00:21:33.570 --> 00:21:35.250
Let's graph it.

00:21:35.250 --> 00:21:35.790
All right.

00:21:35.790 --> 00:21:42.090
Let me draw my axis,
just like that.

00:21:42.090 --> 00:21:46.840
And this is my y-axis, this
is my x-axis, this is y.

00:21:46.840 --> 00:21:49.740
You can even call it the f
of x axis, if you like.

00:21:49.740 --> 00:21:50.820
This is x.

00:21:50.820 --> 00:21:53.750
And so the point 0, 3.29.

00:21:53.750 --> 00:22:01.210
Let's say this is 1, 2, 3,
4, 5, to the point 0, 3.29.

00:22:01.210 --> 00:22:05.290
That's 0, 1, 2, 3, a little bit
above 3, it's right there.

00:22:05.290 --> 00:22:06.950
That's the minimum point.

00:22:06.950 --> 00:22:08.550
And then we're concave.

00:22:08.550 --> 00:22:12.230
The slope is 0 right there, we
figured that out, because the

00:22:12.230 --> 00:22:13.360
first derivative was 0 there.

00:22:13.360 --> 00:22:14.950
So it's a critical
point, and it's concave

00:22:14.950 --> 00:22:15.710
upwards around there.

00:22:15.710 --> 00:22:19.510
So that told us we arepoint at
a minimum point, right there.

00:22:19.510 --> 00:22:22.430
And then at positive 3.

00:22:22.430 --> 00:22:24.400
So 1, 2, 3.

00:22:24.400 --> 00:22:26.950
At positive 3, 4.7.

00:22:26.950 --> 00:22:30.890
So 4.7 will look
something like that.

00:22:30.890 --> 00:22:32.660
We have an inflection point.

00:22:32.660 --> 00:22:36.700
Before that, we're concave
upwards, and then after that

00:22:36.700 --> 00:22:37.460
we're concave downwards.

00:22:37.460 --> 00:22:39.600
So it looks something
like this.

00:22:39.600 --> 00:22:42.520
So we're concave upwards
up with up to that point.

00:22:42.520 --> 00:22:45.820
Maybe, actually, you should,
let me ignore that yellow

00:22:45.820 --> 00:22:47.550
thing I drew before.

00:22:47.550 --> 00:22:49.550
Let me get rid of that.

00:22:49.550 --> 00:22:52.930
Let me draw it like 1, 2, 3.

00:22:52.930 --> 00:22:59.040
3, 4.7 looks like that, and
minus 3, 4.7, 1, 2, 3,

00:22:59.040 --> 00:23:01.290
4.7 looks like that.

00:23:01.290 --> 00:23:07.200
So we know at 0, we are slope
of 0 and we're concave upwards,

00:23:07.200 --> 00:23:09.140
so we look like this.

00:23:09.140 --> 00:23:12.320
We're concave upwards,
until x is equal to 3.

00:23:12.320 --> 00:23:17.690
And at x is equal to 3, we
become concave downwards, and

00:23:17.690 --> 00:23:22.300
we go, let me try my best to
draw it well, and we

00:23:22.300 --> 00:23:23.800
go off like that.

00:23:23.800 --> 00:23:28.170
And then we're concave upwards
around 0, until we get, we're

00:23:28.170 --> 00:23:32.290
concave upwards as long as x is
greater than minus 3, and

00:23:32.290 --> 00:23:34.440
then at minus 3 we become
concave downwards again.

00:23:34.440 --> 00:23:35.620
Maybe I should do
it in that color.

00:23:35.620 --> 00:23:40.010
This concave downwards right
here, that's this, right here.

00:23:40.010 --> 00:23:41.720
That's that, right there.

00:23:41.720 --> 00:23:45.340
And this concave downwards,
right here-- sorry, I meant

00:23:45.340 --> 00:23:48.440
to do it in the red color--
this concave downwards right

00:23:48.440 --> 00:23:51.440
here, is this, right there.

00:23:51.440 --> 00:23:54.490
And then the concave upwards
around 0 is right there.

00:23:54.490 --> 00:23:57.080
You could even imagine, this
concave upwards that we

00:23:57.080 --> 00:23:59.660
measured, that's this, concave
upwards, and then this

00:23:59.660 --> 00:24:01.580
concave upwards is that.

00:24:01.580 --> 00:24:03.700
And then around 0,
we're always upwards.

00:24:03.700 --> 00:24:07.340
So this is my sense of what the
graph will look like and maybe

00:24:07.340 --> 00:24:10.050
it'll just you know it turns
into well you could think about

00:24:10.050 --> 00:24:13.420
what it does is x approaches
positive or negative infinity,

00:24:13.420 --> 00:24:15.230
some of the terms, well,
I won't go into that.

00:24:15.230 --> 00:24:18.430
But let's test whether we've
grafted correctly using

00:24:18.430 --> 00:24:19.570
a graphing calculator.

00:24:19.570 --> 00:24:25.150
So let me get out my TI-85,
trusty TI-85, and let's

00:24:25.150 --> 00:24:26.480
graph this sucker.

00:24:26.480 --> 00:24:28.290
All right, press graph.

00:24:28.290 --> 00:24:38.830
y equals the natural log of
x to the fourth plus 27.

00:24:38.830 --> 00:24:40.660
All right, I want to
hit that graph there.

00:24:40.660 --> 00:24:43.650
So I do second, graph.

00:24:43.650 --> 00:24:45.760
And let's cross our fingers.

00:24:45.760 --> 00:24:47.380
It looks pretty good!

00:24:47.380 --> 00:24:49.770
It looks almost exactly
like what we drew.

00:24:49.770 --> 00:24:53.680
So I think our I think our
mathematics was correct.

00:24:53.680 --> 00:24:55.650
This was actually
very satisfying.

00:24:55.650 --> 00:24:59.060
So hopefully you appreciate the
usefulness of inflection

00:24:59.060 --> 00:25:02.810
points, and second derivative,
and first derivative, in

00:25:02.810 --> 00:25:06.860
graphing some of
these functions.


Graphing using derivatives

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:02.940
Let's see if we can use
everything we know about

00:00:02.940 --> 00:00:06.020
differentiation and
concativity, and maximum and

00:00:06.020 --> 00:00:09.760
minimum points, and inflection
points, to actually graph a

00:00:09.760 --> 00:00:12.530
function without using a
graphing calculator.

00:00:12.530 --> 00:00:16.650
So let's say our function,
let's say that f of x is equal

00:00:16.650 --> 00:00:27.630
to 3x to the fourth minus
4x to the third plus 2.

00:00:27.630 --> 00:00:30.850
And of course, you could always
graph a function just by trying

00:00:30.850 --> 00:00:33.370
out a bunch of points, but we
want to really focus on the

00:00:33.370 --> 00:00:35.830
points that are interesting to
us, and then just to get the

00:00:35.830 --> 00:00:38.240
general shape of the function,
especially we want to focus on

00:00:38.240 --> 00:00:42.160
the things that we can take out
from this function using our

00:00:42.160 --> 00:00:44.930
calculus toolkit, or our
derivative toolkit.

00:00:44.930 --> 00:00:46.940
So the first thing we probably
want to do, is figure

00:00:46.940 --> 00:00:48.680
out the critical points.

00:00:48.680 --> 00:00:54.590
We want to figure out, I'll
write here, critical points.

00:00:54.590 --> 00:00:58.520
And just as a refresher of what
critical points means, it's the

00:00:58.520 --> 00:01:02.320
points where the derivative
of f of x is 0.

00:01:02.320 --> 00:01:06.200
So critical points are f prime
of x is either equal to

00:01:06.200 --> 00:01:08.460
0, or it's undefined.

00:01:12.160 --> 00:01:14.610
This function looks
differentiable everywhere, so

00:01:14.610 --> 00:01:17.320
the critical points that we
worried about are probably,

00:01:17.320 --> 00:01:19.920
well, I can tell you, they're
definitely just the points

00:01:19.920 --> 00:01:22.580
where f prime of x are
going to be equal to 0.

00:01:22.580 --> 00:01:26.620
This derivative, f prime of x,
is going to actually be defined

00:01:26.620 --> 00:01:27.840
over the entire domain.

00:01:27.840 --> 00:01:31.010
So let's actually write down
the derivative right now.

00:01:31.010 --> 00:01:36.390
So the derivative of this,
f prime of x, this is

00:01:36.390 --> 00:01:37.470
pretty straightforward.

00:01:37.470 --> 00:01:42.530
The derivative of 3x to the
fourth, 4 times 3 is 12,

00:01:42.530 --> 00:01:47.420
12x to the, we'll just
decrement the 4 by 1, 3.

00:01:47.420 --> 00:01:47.750
Right?

00:01:47.750 --> 00:01:50.910
You just multiply times the
exponent, and then decrease the

00:01:50.910 --> 00:01:57.020
new exponent by one, minus 3
times 4 is 12, times x to

00:01:57.020 --> 00:01:59.910
the 1 less than 3 is 2.

00:01:59.910 --> 00:02:02.990
And then the derivative of
a constant, the slope of

00:02:02.990 --> 00:02:04.810
a constant, you could
almost imagine, is zero.

00:02:04.810 --> 00:02:05.600
It's not changing.

00:02:05.600 --> 00:02:07.350
A constant, by definition,
isn't changing.

00:02:07.350 --> 00:02:08.884
So that's f prime of x.

00:02:08.884 --> 00:02:11.230
So let's figure out
the critical points.

00:02:11.230 --> 00:02:13.750
The critical points are where
this thing is either going to

00:02:13.750 --> 00:02:16.270
be equal to 0, or
it's undefined.

00:02:16.270 --> 00:02:18.930
Now, I can look over the entire
domain of real numbers, and

00:02:18.930 --> 00:02:20.610
this thing is defined
pretty much anywhere.

00:02:20.610 --> 00:02:22.970
I could put any number here,
and it's not going to blow up.

00:02:22.970 --> 00:02:26.720
It's going to give me an answer
to what the function is.

00:02:26.720 --> 00:02:28.680
So that it's defined
everywhere, so let's

00:02:28.680 --> 00:02:30.400
just figure out where
it's equal to 0.

00:02:30.400 --> 00:02:35.300
So f prime of x is equal to 0.

00:02:35.300 --> 00:02:38.670
So let's solve, which x is,
let's solve-- I don't have

00:02:38.670 --> 00:02:41.130
to rewrite that, I
just wrote that.

00:02:41.130 --> 00:02:43.570
Let's solve for when
this is equal to 0.

00:02:43.570 --> 00:02:45.350
And I'll do it in
the same color.

00:02:45.350 --> 00:02:56.980
So 12x to the third minus
12x squared is equal to 0.

00:02:56.980 --> 00:02:59.530
And so let's what we
can do to solve this.

00:02:59.530 --> 00:03:01.860
We could factor out a 12x.

00:03:01.860 --> 00:03:07.830
So if we factor out a 12x, then
this term becomes just x,

00:03:07.830 --> 00:03:11.450
and then-- actually, let's
factor out a 12x squared.

00:03:11.450 --> 00:03:12.900
We factor out a 12x squared.

00:03:12.900 --> 00:03:15.500
If we divide both of these by
12x squared, this term just

00:03:15.500 --> 00:03:19.990
becomes an x, and then minus
12x squared divided by 12x

00:03:19.990 --> 00:03:22.650
squared is just 1,
is equal to 0.

00:03:22.650 --> 00:03:24.680
I just rewrote this
top thing like this.

00:03:24.680 --> 00:03:25.370
You could go the other way.

00:03:25.370 --> 00:03:28.420
If I distributed this 12x
squared times this entire

00:03:28.420 --> 00:03:31.870
quantity, you would get my
derivative right there.

00:03:31.870 --> 00:03:36.530
So the reason why did that is
because, to solve for 0, or if

00:03:36.530 --> 00:03:41.530
I want all of the x's that make
this equation equal to 0, I now

00:03:41.530 --> 00:03:43.960
have written it in a form
where I'm multiplying one

00:03:43.960 --> 00:03:46.250
thing by another thing.

00:03:46.250 --> 00:03:49.960
And in order for this to be
0, one or both of these

00:03:49.960 --> 00:03:51.480
things must be equal to 0.

00:03:51.480 --> 00:03:55.920
So 12x squared are equal to 0,
which means that x is equal

00:03:55.920 --> 00:03:58.750
to 0 will make this
quantity equals 0.

00:03:58.750 --> 00:04:01.255
And the other thing that would
make this quantity 0 is if

00:04:01.255 --> 00:04:04.100
x minus 1 is equal to 0.

00:04:04.100 --> 00:04:08.050
So x minus 1 is equal to
0 when x is equal to 1.

00:04:08.050 --> 00:04:10.510
So these are 2 critical points.

00:04:10.510 --> 00:04:14.560
Our 2 critical points are x is
equal to 0 and x is equal to 1.

00:04:14.560 --> 00:04:18.190
And remember, those are just
the points where our first

00:04:18.190 --> 00:04:20.050
derivative is equal to 0.

00:04:20.050 --> 00:04:22.280
Where the slope is 0.

00:04:22.280 --> 00:04:25.560
They might be maximum points,
they might be minimum points,

00:04:25.560 --> 00:04:27.400
they might be inflection
points, we don't know.

00:04:27.400 --> 00:04:29.850
They might be, you know, if
this was a constant function,

00:04:29.850 --> 00:04:31.590
they could just be anything.

00:04:31.590 --> 00:04:34.690
So we really can't say a lot
about them just yet, but they

00:04:34.690 --> 00:04:36.300
are points of interest.

00:04:36.300 --> 00:04:37.580
I guess that's all we can say.

00:04:37.580 --> 00:04:39.340
That they are definitely
points of interest.

00:04:39.340 --> 00:04:41.450
But let's keep going, and
let's try to understand the

00:04:41.450 --> 00:04:43.520
concativity, and maybe we can
get a better sense

00:04:43.520 --> 00:04:45.260
of this graph.

00:04:45.260 --> 00:04:49.980
So let's figure out the
second derivative.

00:04:49.980 --> 00:04:53.540
I'll do that in
this orange color.

00:04:53.540 --> 00:04:59.320
So the second derivative of my
function f, let's see, 3 times

00:04:59.320 --> 00:05:05.590
12 is 36x squared minus 24x.

00:05:08.380 --> 00:05:09.220
So let's see.

00:05:09.220 --> 00:05:10.570
Well, there's a couple
of things we can do.

00:05:10.570 --> 00:05:13.290
Now that we know the second
derivative, we can answer the

00:05:13.290 --> 00:05:16.890
question, is my graph concave
upwards or downwards at

00:05:16.890 --> 00:05:18.300
either of these points?

00:05:18.300 --> 00:05:20.270
So let's figure out what,
at either of these

00:05:20.270 --> 00:05:21.780
critical points.

00:05:21.780 --> 00:05:23.010
And it'll all fit together.

00:05:23.010 --> 00:05:25.140
Remember, if it's concave
upwards, then we're

00:05:25.140 --> 00:05:26.340
kind of in a U shape.

00:05:26.340 --> 00:05:29.370
If it's concave downwards,
then we're in a kind of

00:05:29.370 --> 00:05:30.850
upside down U shape.

00:05:30.850 --> 00:05:34.400
So f prime prime, our second
derivative, at x is equal

00:05:34.400 --> 00:05:36.800
to 0, is equal to what?

00:05:36.800 --> 00:05:40.530
It's equal to 36 0 squared
minus 24 times 0.

00:05:40.530 --> 00:05:42.350
So that's just 0.

00:05:42.350 --> 00:05:45.210
So f prime prime is
just equal to 0.

00:05:45.210 --> 00:05:49.120
We're neither concave upwards
nor concave downwards here.

00:05:49.120 --> 00:05:50.420
It might be a transition point.

00:05:50.420 --> 00:05:50.990
It may not.

00:05:50.990 --> 00:05:52.850
If it is a transition point,
then we're dealing with

00:05:52.850 --> 00:05:53.620
an inflection point.

00:05:53.620 --> 00:05:55.190
We're not sure yet.

00:05:55.190 --> 00:05:57.910
Now let's see what f prime
prime, our second derivative,

00:05:57.910 --> 00:06:00.190
evaluated at 1 is.

00:06:00.190 --> 00:06:03.580
So that's 36 times 1, let me
write it down, that's equal

00:06:03.580 --> 00:06:11.910
to 36 times 1 squared, which
is 36, minus 24 times 1.

00:06:11.910 --> 00:06:16.650
So it's 36 minus 24,
so it's equal to 12.

00:06:16.650 --> 00:06:20.150
So this is positive, our second
derivative is positive here.

00:06:20.150 --> 00:06:22.880
It's equal to 12, which
means our first derivative,

00:06:22.880 --> 00:06:24.500
our slope is increasing.

00:06:24.500 --> 00:06:26.950
The rate of change of our
slope is positive here.

00:06:26.950 --> 00:06:30.945
So at this point right here,
we are concave upwards.

00:06:34.020 --> 00:06:36.200
Which tells me that this
is probably a minimum

00:06:36.200 --> 00:06:36.770
point, right?

00:06:36.770 --> 00:06:40.550
The slope is 0 here, but we are
concave upwards at that point.

00:06:40.550 --> 00:06:41.880
So that's interesting.

00:06:41.880 --> 00:06:43.920
So let's see if there
any other potential

00:06:43.920 --> 00:06:44.780
inflection points here.

00:06:44.780 --> 00:06:49.460
We already know that this is a
potential inflection point.

00:06:49.460 --> 00:06:50.680
Let me circle it in red.

00:06:50.680 --> 00:06:52.640
It's a potential
inflection point.

00:06:52.640 --> 00:06:54.830
We don't know whether
our function actually

00:06:54.830 --> 00:06:55.970
transitions at that point.

00:06:55.970 --> 00:06:58.310
We'll have to experiment
a little bit to see if

00:06:58.310 --> 00:06:59.130
that's really the case.

00:06:59.130 --> 00:07:02.860
But let's see if there any
other inflection points, or

00:07:02.860 --> 00:07:04.140
potential inflection points.

00:07:04.140 --> 00:07:07.050
So let's see if this
equals 0 anywhere else.

00:07:07.050 --> 00:07:11.790
So 36 x squared minus
24 x is equal to 0.

00:07:11.790 --> 00:07:13.240
Let's solve for x.

00:07:13.240 --> 00:07:17.780
Let us factor out, well,
we can factor out 12x.

00:07:17.780 --> 00:07:25.070
12x times 3x, right, 3x
times 12x is 36x squared,

00:07:25.070 --> 00:07:28.560
minus 2, is equal to 0.

00:07:28.560 --> 00:07:31.090
So these two are
equivalent expressions.

00:07:31.090 --> 00:07:33.520
If you multiply this out,
you'll get this thing up here.

00:07:33.520 --> 00:07:36.510
So this thing is going to be
equal to 0, either if 12 x is

00:07:36.510 --> 00:07:39.710
equal to 0, so 12 x is equal
to zero, that gives us

00:07:39.710 --> 00:07:41.240
x is equal to zero.

00:07:41.240 --> 00:07:43.880
So at x equals 0,
this thing equals 0.

00:07:43.880 --> 00:07:46.850
So the second derivative is 0
there, and we already knew

00:07:46.850 --> 00:07:48.690
that, because we tested
that number out.

00:07:48.690 --> 00:07:52.180
Or this thing, if this
expression was 0, then the

00:07:52.180 --> 00:07:54.970
entire second derivative
would also be zero.

00:07:54.970 --> 00:07:55.740
So let's write that.

00:07:55.740 --> 00:08:03.090
So 3x minus 2 is equal to 0, 3x
is equal to 2, just adding 2 to

00:08:03.090 --> 00:08:07.280
both sides, 3x is equal to 2/3.

00:08:07.280 --> 00:08:11.310
So this is another interesting
point that we haven't really

00:08:11.310 --> 00:08:14.050
hit upon before that might
be an inflection point.

00:08:14.050 --> 00:08:15.620
The reason why is it
might be, is because the

00:08:15.620 --> 00:08:18.330
second derivative is
definitely 0 here.

00:08:18.330 --> 00:08:20.580
You put 2/3 here,
you're going to get 0.

00:08:20.580 --> 00:08:22.830
So what we have to do, is see
whether the second derivative

00:08:22.830 --> 00:08:28.050
is positive or negative
on either side of 2/3.

00:08:28.050 --> 00:08:29.200
We already have a
sense of that.

00:08:29.200 --> 00:08:31.540
I mean, we could try out
a couple of numbers.

00:08:31.540 --> 00:08:37.210
We know that, you know, if we
say that x is greater than 2/3.

00:08:37.210 --> 00:08:38.690
Let me scroll down a
little bit, just so

00:08:38.690 --> 00:08:40.420
we have some space.

00:08:40.420 --> 00:08:50.410
So let's see what happens
when x is greater than 2/3,

00:08:50.410 --> 00:08:51.580
what is f prime prime?

00:08:51.580 --> 00:08:53.470
What is the second derivative?

00:08:53.470 --> 00:08:55.900
So let's try out a value
that's pretty close, just

00:08:55.900 --> 00:08:59.020
to get a sense of things.

00:08:59.020 --> 00:09:04.190
So let me rewrite it. f prime
prime of x is equal to,

00:09:04.190 --> 00:09:05.080
let me write like this.

00:09:05.080 --> 00:09:06.550
I mean, I could write like
that, but this might be

00:09:06.550 --> 00:09:07.490
easier to deal with.

00:09:07.490 --> 00:09:13.030
It's equal to 12x
times 3x minus 2.

00:09:13.030 --> 00:09:16.790
So if x is greater than 2/3,
this term right here is

00:09:16.790 --> 00:09:18.270
going to be positive.

00:09:18.270 --> 00:09:21.240
That's definitely, any
positive number times 12

00:09:21.240 --> 00:09:22.560
is going to be positive.

00:09:22.560 --> 00:09:25.390
But what about this
term, right here?

00:09:25.390 --> 00:09:29.270
3 times 2/3 minus 2
is exactly 0, right?

00:09:29.270 --> 00:09:30.490
That's 2 minus 2.

00:09:30.490 --> 00:09:33.270
But anything larger than that,
3 times, you know, if I had

00:09:33.270 --> 00:09:38.670
2.1/3, this is going to
be a positive quantity.

00:09:38.670 --> 00:09:41.670
Any value of x greater than
2/3 will make this thing

00:09:41.670 --> 00:09:43.580
right here positive.

00:09:43.580 --> 00:09:44.400
Right?

00:09:44.400 --> 00:09:46.670
This thing is also
going to be positive.

00:09:46.670 --> 00:09:50.250
So that means that when x
is greater than 2/3, that

00:09:50.250 --> 00:09:54.700
tells us that the second
derivative is positive.

00:09:54.700 --> 00:09:56.800
It is greater than 0.

00:09:56.800 --> 00:10:00.530
So in our domain, as long
as x is larger than 2/3,

00:10:00.530 --> 00:10:02.110
we are concave upwards.

00:10:02.110 --> 00:10:05.200
And we saw that here,
at x is equal to 1.

00:10:05.200 --> 00:10:06.840
We were concave upwards.

00:10:06.840 --> 00:10:10.240
But what about x
being less than 2/3?

00:10:10.240 --> 00:10:13.540
So when x is less than 2/3,
let me write it, let me

00:10:13.540 --> 00:10:15.720
scroll down a little bit.

00:10:15.720 --> 00:10:19.170
When x is less than
2/3, what's going on?

00:10:19.170 --> 00:10:20.320
I'll rewrite it.

00:10:20.320 --> 00:10:22.500
f prime prime of x,
second derivative,

00:10:22.500 --> 00:10:27.000
12x times 3x minus 2.

00:10:27.000 --> 00:10:30.020
Well, if we go really far left,
we're going to get a negative

00:10:30.020 --> 00:10:31.230
number here, and this might
be negative.

00:10:31.230 --> 00:10:35.020
But if we just go right
below 2/3, when we're still

00:10:35.020 --> 00:10:36.700
in the positive domain.

00:10:36.700 --> 00:10:41.250
So if this was like 1.9/3,
which is a mix of a decimal and

00:10:41.250 --> 00:10:43.640
a fraction, or even 1/3, this
thing is still going

00:10:43.640 --> 00:10:44.810
to be positive.

00:10:44.810 --> 00:10:47.910
Right below 2/3, this thing is
still going to be positive.

00:10:47.910 --> 00:10:50.265
We're going to be multiplying
12 by a positive number.

00:10:50.265 --> 00:10:53.310
But what's going on right here?

00:10:55.850 --> 00:10:57.760
At 2/3, we're exactly 0.

00:10:57.760 --> 00:11:00.700
But as you go to anything
less than 2/3, 3

00:11:00.700 --> 00:11:02.230
times 1/3 is only 1.

00:11:02.230 --> 00:11:04.820
1 minus 2, you're going
to get negative numbers.

00:11:04.820 --> 00:11:07.280
So when x is less than 2/3,
this thing right here is

00:11:07.280 --> 00:11:09.290
going to be negative.

00:11:09.290 --> 00:11:13.480
So the second derivative, if x
is less than 2/3, the second

00:11:13.480 --> 00:11:16.370
derivative, right to the left,
right when you go less than

00:11:16.370 --> 00:11:20.270
2/3, the seconds derivative
of x is less than 0.

00:11:20.270 --> 00:11:23.690
Now the fact that we have this
transition, from when we're

00:11:23.690 --> 00:11:28.115
less than 2/3, we have a
negative second derivative, and

00:11:28.115 --> 00:11:30.340
when we're greater than 2/3, we
have a positive second

00:11:30.340 --> 00:11:34.300
derivative, that tells us that
this, indeed, is an

00:11:34.300 --> 00:11:36.800
inflection point.

00:11:36.800 --> 00:11:40.240
That x is equal to 2/3 thirds
is definitely an inflection

00:11:40.240 --> 00:11:43.940
point for our original
function up here.

00:11:43.940 --> 00:11:47.180
Now, we have one more candidate
inflection point, and then

00:11:47.180 --> 00:11:48.210
we're ready to graph.

00:11:48.210 --> 00:11:49.970
Then, you know, once you do all
the inflection points and the

00:11:49.970 --> 00:11:53.700
max and the minimum, you are
ready to graph the function.

00:11:53.700 --> 00:11:56.460
So let's see if x is equal to
0 is an inflection point.

00:11:56.460 --> 00:11:59.520
We know that the second
derivative is 0 at 0.

00:11:59.520 --> 00:12:02.380
But what happens above and
below the second derivative?

00:12:02.380 --> 00:12:04.670
So let me do our
little test here.

00:12:04.670 --> 00:12:09.310
So when x is, let me draw a
line so we don't get confused

00:12:09.310 --> 00:12:12.070
with all of the stuff
that I wrote here.

00:12:12.070 --> 00:12:16.400
So when x is greater than
0, what's happening in

00:12:16.400 --> 00:12:17.840
the second derivative?

00:12:17.840 --> 00:12:21.850
Remember, the second
derivative was equal to

00:12:21.850 --> 00:12:24.790
12x times 3x minus 2.

00:12:24.790 --> 00:12:27.010
I like writing it this way,
because you've kind of

00:12:27.010 --> 00:12:29.830
decomposed it into two linear
expressions, and you could see

00:12:29.830 --> 00:12:32.030
whether each of them are
positive or negative.

00:12:32.030 --> 00:12:35.720
So if x is greater than 0, this
thing right here is definitely

00:12:35.720 --> 00:12:39.900
going to be positive, and then
this thing right here, right

00:12:39.900 --> 00:12:42.820
when you go right above x is
greater than 0, so we have to

00:12:42.820 --> 00:12:45.810
make sure to be very close
to this number, right?

00:12:45.810 --> 00:12:50.360
So this number,
let's say it's 0.1.

00:12:50.360 --> 00:12:52.070
You're right above 0.

00:12:52.070 --> 00:12:54.260
So this isn't going to be true
for all of x greater than 0.

00:12:54.260 --> 00:12:56.000
We just want to test exactly
what happens, right when

00:12:56.000 --> 00:12:57.920
we go right above 0.

00:12:57.920 --> 00:12:59.020
So this is 0.1.

00:12:59.020 --> 00:13:02.080
You would have 0.3, 0.3
minus 2, that would be a

00:13:02.080 --> 00:13:03.520
negative number, right?

00:13:03.520 --> 00:13:06.470
So right as x goes right
above 0, this thing

00:13:06.470 --> 00:13:08.410
right here is negative.

00:13:08.410 --> 00:13:12.630
So at x is greater than 0,
you will have your second

00:13:12.630 --> 00:13:15.250
derivative is going
to be less than 0.

00:13:15.250 --> 00:13:17.010
You're concave downwards.

00:13:17.010 --> 00:13:19.530
Which makes sense, because at
some point, we're going to

00:13:19.530 --> 00:13:20.900
be hitting a transition.

00:13:20.900 --> 00:13:23.430
Remember, we were concave
downwards before we

00:13:23.430 --> 00:13:25.250
got to 2/3, right?

00:13:25.250 --> 00:13:26.400
So this is consistent.

00:13:26.400 --> 00:13:30.580
From 0 to 2/3, we are concave
downwards, and then at 2/3,

00:13:30.580 --> 00:13:32.540
we become concave upwards.

00:13:32.540 --> 00:13:38.390
Now let's see what happens when
x is right less than, when x

00:13:38.390 --> 00:13:42.400
is just barely, just
barely less than 0.

00:13:42.400 --> 00:13:45.520
So once again, f prime, the
second derivitive of x is equal

00:13:45.520 --> 00:13:49.030
to 12x times 3x minus 2.

00:13:49.030 --> 00:13:49.590
Well, right.

00:13:49.590 --> 00:13:54.830
If x was minus 0.1 or 0.0001,
no matter what, this thing is

00:13:54.830 --> 00:13:57.470
going to be negative, this
expression right here is going

00:13:57.470 --> 00:14:01.410
to be negative, the 12x, right,
you just have some negative

00:14:01.410 --> 00:14:03.910
value here, times 12, is
going to be negative.

00:14:03.910 --> 00:14:05.940
And then what's
this going to be?

00:14:05.940 --> 00:14:10.000
Well, 3 times minus 0.1 is
going to be minus 0.3,

00:14:10.000 --> 00:14:12.090
minus 2 is minus 2.3.

00:14:12.090 --> 00:14:13.250
You're definitely going
to have a negative.

00:14:13.250 --> 00:14:15.363
This value right here is going
to be negative, and then when

00:14:15.363 --> 00:14:17.075
you subtract from a negative,
it's definitely going

00:14:17.075 --> 00:14:18.120
to be negative.

00:14:18.120 --> 00:14:20.270
So that is also going
to be negative.

00:14:20.270 --> 00:14:22.090
But if you multiply a negative
times a negative, you're

00:14:22.090 --> 00:14:23.290
going to get a positive.

00:14:23.290 --> 00:14:27.660
So actually, right below x
is less than 0, the second

00:14:27.660 --> 00:14:28.455
derivative is positive.

00:14:32.420 --> 00:14:35.140
Now, this all might have been a
little bit confusing, but we

00:14:35.140 --> 00:14:37.120
should now have the payoff.

00:14:37.120 --> 00:14:38.390
We now have the payoff.

00:14:38.390 --> 00:14:41.050
We have all of the
interesting things going on.

00:14:41.050 --> 00:14:46.060
We know that at x is equal to
1, we know that at x is equal

00:14:46.060 --> 00:14:48.690
to 1, let me write
it over here.

00:14:48.690 --> 00:14:55.106
We've figured out at x is
equal to 1, the slope is 0.

00:14:55.106 --> 00:14:59.840
So f prime prime is, sorry,
let me write this way.

00:14:59.840 --> 00:15:03.500
I should have said, we
know that the slope is 0.

00:15:03.500 --> 00:15:04.670
Slope is equal to 0.

00:15:04.670 --> 00:15:07.020
And we figured that out because
the first derivative was 0.

00:15:07.020 --> 00:15:09.000
This was a critical point.

00:15:09.000 --> 00:15:12.860
And we know that we're dealing
with, the function is concave

00:15:12.860 --> 00:15:14.860
upwards at this point.

00:15:14.860 --> 00:15:17.740
And that tells us that this is
going to be a minimum point.

00:15:21.100 --> 00:15:22.520
And we should actually get
the coordinates so we

00:15:22.520 --> 00:15:23.390
can actually graph it.

00:15:23.390 --> 00:15:24.980
That was the whole
point of this video.

00:15:24.980 --> 00:15:29.600
So f of 1 is equal to what?

00:15:29.600 --> 00:15:33.510
f of 1, let's go back to our
original function, is 3 times

00:15:33.510 --> 00:15:38.190
1, right, 1 to the fourth is
just 1, 3 times 1 minus

00:15:38.190 --> 00:15:40.810
4 plus 2, right?

00:15:40.810 --> 00:15:47.510
So it's 3 times 1 minus 4 times
1, which is minus 1, plus 2,

00:15:47.510 --> 00:15:48.930
well, that's just a positive 1.

00:15:48.930 --> 00:15:50.920
So f of 1 is one.

00:15:50.920 --> 00:15:56.420
And then we know at x is equal
to 0, we also figured out that

00:15:56.420 --> 00:15:59.260
the slope is equal to 0.

00:15:59.260 --> 00:16:02.020
But we figured out that this
was an inflection point, right?

00:16:02.020 --> 00:16:05.040
The concativity switches
before and after.

00:16:05.040 --> 00:16:06.510
So this is an inflection point.

00:16:10.080 --> 00:16:15.700
And we are concave below
0, so when x is less

00:16:15.700 --> 00:16:17.535
than 0, we are upwards.

00:16:20.040 --> 00:16:22.180
Our second derivative
is positive.

00:16:22.180 --> 00:16:25.250
And when x is it greater
than 0, we are downwards.

00:16:25.250 --> 00:16:26.420
We're concave downwards.

00:16:26.420 --> 00:16:28.400
Right above, not for all
of the [? domain ?]

00:16:28.400 --> 00:16:32.760
x and 0, just right
above 0, downwards.

00:16:32.760 --> 00:16:35.520
And then what is f of 0,
just so we know, because we

00:16:35.520 --> 00:16:37.660
want to graph that point?

00:16:37.660 --> 00:16:39.960
f of 0.

00:16:39.960 --> 00:16:41.620
See, f of 0, this is easy.

00:16:41.620 --> 00:16:45.380
3 times 0 minus 4 times 0
plus 2, that's just 2.

00:16:45.380 --> 00:16:48.080
f of 0 is 2.

00:16:48.080 --> 00:16:50.630
And then finally we got the
point, x is equal to 2/3.

00:16:53.270 --> 00:16:56.420
Let me do that in
another color.

00:16:56.420 --> 00:17:00.970
We had the point x
is equal to 2/3.

00:17:00.970 --> 00:17:05.670
We figured out that this
was an inflection point.

00:17:05.670 --> 00:17:09.670
The slope definitely isn't 0
there, because it wasn't one

00:17:09.670 --> 00:17:11.740
of the critical points.

00:17:11.740 --> 00:17:15.390
And we know that
we are downwards.

00:17:15.390 --> 00:17:19.210
We know that when x is less
than 2/3, or right less than

00:17:19.210 --> 00:17:22.100
2/3, we are concave downwards.

00:17:22.100 --> 00:17:27.120
And when x is greater than 2/3,
we saw it up here, when x was

00:17:27.120 --> 00:17:31.810
greater than 2/3, right up
here, we were concave upwards.

00:17:31.810 --> 00:17:35.040
The second derivative
was positive.

00:17:35.040 --> 00:17:35.885
We were upwards.

00:17:38.460 --> 00:17:40.790
Now we could actually figure
out, what's f of 2/3?

00:17:40.790 --> 00:17:43.420
That's actually a little
bit complicated.

00:17:43.420 --> 00:17:44.530
We don't even have to
figure that out, I

00:17:44.530 --> 00:17:45.320
don't think, to graph.

00:17:45.320 --> 00:17:48.180
I think we could do a pretty
good job of graphing it just

00:17:48.180 --> 00:17:49.270
with what we know right now.

00:17:49.270 --> 00:17:50.370
So that's our take aways.

00:17:50.370 --> 00:17:52.740
Let me do a rough graph.

00:17:52.740 --> 00:17:54.340
Let's see.

00:17:54.340 --> 00:17:57.840
So let me do my axes,
just like that.

00:17:57.840 --> 00:18:02.200
So we're going to want to
graph the point 0, 2.

00:18:02.200 --> 00:18:04.120
So let's say that
the point, 0, 2.

00:18:04.120 --> 00:18:08.570
So this is x is equal to
0, and we go up, 1, 2.

00:18:08.570 --> 00:18:10.610
So this is the point 0, 2.

00:18:10.610 --> 00:18:12.640
Maybe I'll do it in that
color, the color I was

00:18:12.640 --> 00:18:14.480
using, so that's this color.

00:18:14.480 --> 00:18:17.050
So that's that
point right there.

00:18:17.050 --> 00:18:19.650
Then we have the point x,
we have f of 1, which is

00:18:19.650 --> 00:18:21.930
the point 1, 1, right?

00:18:21.930 --> 00:18:23.090
So this point right here.

00:18:26.100 --> 00:18:29.150
So that's the point 1, 1.

00:18:29.150 --> 00:18:32.250
This was the point 0, 2.

00:18:32.250 --> 00:18:36.260
And then we have x is equal
to 2/3, which is our

00:18:36.260 --> 00:18:36.940
inflection point.

00:18:36.940 --> 00:18:39.730
So when x is 2/3, we
don't know exactly what

00:18:39.730 --> 00:18:41.110
number f of 2/3 is.

00:18:41.110 --> 00:18:42.350
Maybe here someplace.

00:18:42.350 --> 00:18:43.890
Let's say f of 2/3
is right there.

00:18:43.890 --> 00:18:48.840
So that's the point, 2/3, and
then wherever f of 2/3 is.

00:18:48.840 --> 00:18:50.960
It looks like it's going
to be 1 point something.

00:18:50.960 --> 00:18:52.280
f of 2/3.

00:18:52.280 --> 00:18:53.840
You could calculate it, if
you like, you just have

00:18:53.840 --> 00:18:55.500
to substitute back
in the function.

00:18:55.500 --> 00:18:59.440
But we're ready to
graph this thing.

00:18:59.440 --> 00:19:04.100
So we know that at x is
equal to 1, the slope is 0.

00:19:04.100 --> 00:19:05.110
We know that the slope is 0.

00:19:05.110 --> 00:19:05.750
It's flat here.

00:19:05.750 --> 00:19:07.520
We know it's concave upwards.

00:19:07.520 --> 00:19:12.070
So we're dealing, it looks
like this, looks like

00:19:12.070 --> 00:19:13.030
that over that interval.

00:19:13.030 --> 00:19:14.480
We're concave upwards.

00:19:14.480 --> 00:19:17.440
And we know we're concave
upwards from x is equal

00:19:17.440 --> 00:19:19.480
to 2/3 and on, right?

00:19:19.480 --> 00:19:20.690
Let me do it in that color.

00:19:20.690 --> 00:19:23.310
We knew x equals 2/3 and
on, we're concave upwards.

00:19:23.310 --> 00:19:25.670
And that's why I was able
to draw this U-shape.

00:19:25.670 --> 00:19:29.390
Now we know that when x is less
than 2/3 and greater than

00:19:29.390 --> 00:19:31.760
0, we're concave downwards.

00:19:31.760 --> 00:19:33.740
So the graph would look
something like this,

00:19:33.740 --> 00:19:34.270
over this interval.

00:19:34.270 --> 00:19:36.890
We'll be concave downwards.

00:19:36.890 --> 00:19:38.820
Let me draw it nicely.

00:19:38.820 --> 00:19:41.080
Over this interval, the
slope is decreasing.

00:19:41.080 --> 00:19:42.950
And you could see it, if you
keep drawing tangent lines.

00:19:42.950 --> 00:19:45.660
It's flattish there, it gets
negative, more negative, more

00:19:45.660 --> 00:19:48.490
negative, until the inflection
point, and then it starts

00:19:48.490 --> 00:19:51.330
increasing again, because we
go back to concave upwards.

00:19:51.330 --> 00:19:54.600
And then finally, the last
interval is below 0, and we

00:19:54.600 --> 00:19:58.280
know below 0, when x is less
than 0, we're concave upwards.

00:19:58.280 --> 00:20:01.110
So the graph looks like this.

00:20:01.110 --> 00:20:02.820
The graph looks like that.

00:20:02.820 --> 00:20:05.490
And we also know that x is
equal to 0 was a critical

00:20:05.490 --> 00:20:06.620
point, the slope of 0.

00:20:06.620 --> 00:20:08.820
So the graph is actually
flat right there, too.

00:20:08.820 --> 00:20:11.880
So this is an inflection point
where the slope was also 0.

00:20:11.880 --> 00:20:13.260
So this is our final graph.

00:20:13.260 --> 00:20:14.060
We're done.

00:20:14.060 --> 00:20:16.550
After all that work, we were
able to use our calculus

00:20:16.550 --> 00:20:18.590
skills, and our knowledge of
inflection points, and

00:20:18.590 --> 00:20:21.600
concativity, and transitions
in concativity, to

00:20:21.600 --> 00:20:25.510
actually graph this fairly
hairy-looking graph.

00:20:25.510 --> 00:20:28.395
But this should be kind of what
it looks like, if you graph

00:20:28.395 --> 00:20:30.100
it on your calculator.


Inflection Points and Concavity Intuition

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:03.770
The idea of concativity
will show up a lot in

00:00:03.770 --> 00:00:04.800
your calculus class.

00:00:04.800 --> 00:00:11.110
Well, what I want to do in this
video is, one, show you what

00:00:11.110 --> 00:00:14.440
concativity is, or what concave
upwards and what concave

00:00:14.440 --> 00:00:18.280
downwards is, and to have an
intuition of what those mean,

00:00:18.280 --> 00:00:21.090
and then discuss the ideas of
inflection points, which are

00:00:21.090 --> 00:00:24.690
really just transition points
between being either concave

00:00:24.690 --> 00:00:28.090
upwards and downwards or
between downwards and upwards.

00:00:28.090 --> 00:00:31.670
So the two ideas that we'll
talk about, one is being

00:00:31.670 --> 00:00:38.460
concave upwards-- I'll write
that in this column right

00:00:38.460 --> 00:00:42.710
here, and maybe in pink I'll
show you what the concave

00:00:42.710 --> 00:00:43.900
downwards looks like.

00:00:48.720 --> 00:00:52.260
So a very simple way to think
about it is that concave

00:00:52.260 --> 00:00:54.450
upwards is kind of a U shape.

00:00:54.450 --> 00:00:56.080
That's why it's called
concave upwards.

00:00:56.080 --> 00:00:58.230
It might look
something like this.

00:00:58.230 --> 00:01:00.090
Let me draw some axes so
you know that I'm actually

00:01:00.090 --> 00:01:02.740
graphing something and
not just drawing a U.

00:01:02.740 --> 00:01:07.330
So maybe if that is my axis
right there, a concave

00:01:07.330 --> 00:01:09.320
upwards graph would look
something like this.

00:01:09.320 --> 00:01:12.590
And we do it in green.

00:01:12.590 --> 00:01:15.140
It could look
something like this.

00:01:15.140 --> 00:01:19.030
So this graph, it'll just
keep going as x goes

00:01:19.030 --> 00:01:20.050
positive and negative.

00:01:20.050 --> 00:01:24.160
Over the entire domain of this
function, this is my f of x

00:01:24.160 --> 00:01:27.000
right here, it is concave
upwards, and you can see

00:01:27.000 --> 00:01:28.260
it has this U shape.

00:01:28.260 --> 00:01:31.840
And I'll discuss in a second
what implication that has for

00:01:31.840 --> 00:01:35.510
its slope or what the slope is
doing, but this is just very

00:01:35.510 --> 00:01:37.580
easy to recognize visually.

00:01:37.580 --> 00:01:40.680
Similarly, let me
draw an axis again.

00:01:40.680 --> 00:01:43.775
I want to show you what
concave downwards looks like.

00:01:43.775 --> 00:01:46.460
Let me draw some
axes right there.

00:01:46.460 --> 00:01:48.770
I'm just trying an
arbitrary function f of x.

00:01:48.770 --> 00:01:49.470
It could be anywhere.

00:01:49.470 --> 00:01:52.140
It doesn't have to be-- you
know, the bottom point doesn't

00:01:52.140 --> 00:01:53.640
have to be in the first
quadrant like this.

00:01:53.640 --> 00:01:55.500
It's just the general U
shape I wanted to show

00:01:55.500 --> 00:01:57.070
you for concave upwards.

00:01:57.070 --> 00:01:59.410
Concave downwards, you could
probably guess what it looks

00:01:59.410 --> 00:02:01.580
like based on-- if this is
upwards, downwards is just

00:02:01.580 --> 00:02:02.130
going to be opposite.

00:02:02.130 --> 00:02:04.866
It might look
something like this.

00:02:04.866 --> 00:02:07.170
It might look something like
this, so maybe this is some

00:02:07.170 --> 00:02:09.490
other function g of x.

00:02:09.490 --> 00:02:11.680
And notice, it looks
like an upside down U.

00:02:11.680 --> 00:02:14.120
Over the entire domain
of this function, it

00:02:14.120 --> 00:02:15.480
is concave downwards.

00:02:15.480 --> 00:02:18.490
So you can look at it, and
immediately see that, oh, well,

00:02:18.490 --> 00:02:20.560
if it's like a U, it's upwards.

00:02:20.560 --> 00:02:22.635
If it's like an upside
down U, it's downwards.

00:02:22.635 --> 00:02:25.100
But what does that actually
mean for the slope?

00:02:25.100 --> 00:02:27.950
So to understand that, let's
think about what's happening

00:02:27.950 --> 00:02:29.180
to the slope here.

00:02:29.180 --> 00:02:30.420
So I'm just going
to do it visually.

00:02:30.420 --> 00:02:33.480
So at that point, the slope of
the tangent line, or the slope

00:02:33.480 --> 00:02:36.400
of the actual graph, or the
instantaneous slope, however

00:02:36.400 --> 00:02:40.020
you want to view it-- let me
see how well I can draw it--

00:02:40.020 --> 00:02:43.690
it'll look something like that,
and that's the tangent line.

00:02:43.690 --> 00:02:47.420
Then as we increase x, so
that's at this x-value

00:02:47.420 --> 00:02:48.500
right here, right?

00:02:48.500 --> 00:02:50.460
At that x-value, the
slope looks like that.

00:02:50.460 --> 00:02:52.240
It's fairly negative.

00:02:52.240 --> 00:02:54.800
Now, as we increase x, maybe
we go to another point

00:02:54.800 --> 00:02:56.370
right here, what happens?

00:02:56.370 --> 00:02:59.265
The tangent line will
look like this.

00:03:01.860 --> 00:03:07.180
It's still downward sloping,
but it's less downward

00:03:07.180 --> 00:03:08.220
sloping now, right?

00:03:08.220 --> 00:03:10.960
Here, it was a very
steep downward slope.

00:03:10.960 --> 00:03:13.520
Now it's a less steep
downward slope.

00:03:13.520 --> 00:03:16.920
And we can keep going if
we go to this point.

00:03:16.920 --> 00:03:21.590
We're still downward sloping,
but even less steep.

00:03:21.590 --> 00:03:22.840
And that keeps happening.

00:03:22.840 --> 00:03:27.740
We keep getting less and less
downward sloping, or less steep

00:03:27.740 --> 00:03:30.590
in the downward direction,
until we get to this minimum

00:03:30.590 --> 00:03:33.660
point, and you've seen this
before, where our slope

00:03:33.660 --> 00:03:37.230
actually gets to zero, and
it just keeps increasing.

00:03:37.230 --> 00:03:38.340
It keeps increasing over here.

00:03:38.340 --> 00:03:41.350
If we go to this point, now
we're going-- we have a

00:03:41.350 --> 00:03:42.630
steep upward increase.

00:03:42.630 --> 00:03:45.260
As we increase our x even
further, the slope at this

00:03:45.260 --> 00:03:49.590
point right here, it's
increased even more.

00:03:49.590 --> 00:03:53.620
It's a steeper upward
sloping curve.

00:03:53.620 --> 00:03:56.520
So I want to make it very
clear, even though if we go to

00:03:56.520 --> 00:03:59.410
this left-hand point here, we
had a very steep downward

00:03:59.410 --> 00:04:02.850
sloping curve, and as we went
up here, we went upward sloping

00:04:02.850 --> 00:04:06.820
the whole time as our x
increased, the whole time as

00:04:06.820 --> 00:04:13.290
we increased x, what was
happening to the slope?

00:04:13.290 --> 00:04:15.450
The slope was increasing.

00:04:15.450 --> 00:04:20.950
So as we increased x,
the slope increased.

00:04:20.950 --> 00:04:26.270
And this is the definition of
a concave upwards interval

00:04:26.270 --> 00:04:27.340
or section of this curve.

00:04:27.340 --> 00:04:29.590
In this case, it was
this entire curve.

00:04:29.590 --> 00:04:31.580
I'll start showing you
functions that mix it

00:04:31.580 --> 00:04:32.670
up in a little bit.

00:04:32.670 --> 00:04:34.630
Over here, we were
very negative.

00:04:34.630 --> 00:04:35.970
We became less negative.

00:04:35.970 --> 00:04:38.530
Becoming less negative is the
same thing as increasing.

00:04:38.530 --> 00:04:40.640
Even less negative over here.

00:04:40.640 --> 00:04:42.090
Then we were completely flat.

00:04:42.090 --> 00:04:45.490
Zero is a higher slope than
some negative slope here.

00:04:45.490 --> 00:04:48.760
And then we became positive,
and more positive, and even

00:04:48.760 --> 00:04:52.550
more positive, so the entire
time our slope is increasing.

00:04:52.550 --> 00:04:55.820
Now, let's look at the
concave downward scenario.

00:04:55.820 --> 00:04:58.710
Let's start with a relatively
low x, maybe right here.

00:04:58.710 --> 00:05:02.340
Here we have a pretty high
slope, right, at this point?

00:05:02.340 --> 00:05:05.060
We have a very steep
upward slope.

00:05:05.060 --> 00:05:09.320
As x increases, so we go to
a less negative x, so x is

00:05:09.320 --> 00:05:11.740
increasing, we still have an
upward slope, but

00:05:11.740 --> 00:05:13.050
it's less upward.

00:05:13.050 --> 00:05:15.860
So our slope has decreased.

00:05:15.860 --> 00:05:20.180
And over here, it's still an
upward sloping curve, right?

00:05:20.180 --> 00:05:22.180
It's going from the bottom
left to the top right,

00:05:22.180 --> 00:05:23.790
but it's flatter.

00:05:23.790 --> 00:05:24.510
It's flattening.

00:05:24.510 --> 00:05:27.060
It's becoming less positive
until we get to this point.

00:05:27.060 --> 00:05:29.550
I didn't draw the original
curve that well, where we get

00:05:29.550 --> 00:05:36.840
to maybe a maximum point
because the slope goes to zero.

00:05:36.840 --> 00:05:39.880
And as we increase now, our
slope becomes negative.

00:05:39.880 --> 00:05:41.930
It becomes even more negative.

00:05:41.930 --> 00:05:45.950
It becomes even more negative,
so as we kind of travel, as we

00:05:45.950 --> 00:05:53.310
increase our x, as we increase
our x over this curve,

00:05:53.310 --> 00:05:56.010
our slope decreases.

00:05:56.010 --> 00:05:59.150
Slope is decreasing.

00:05:59.150 --> 00:06:00.105
Slope decreases.

00:06:00.105 --> 00:06:01.310
I want to make that very clear.

00:06:01.310 --> 00:06:02.600
The slope was very positive.

00:06:02.600 --> 00:06:04.740
This doesn't mean that
the slope is negative.

00:06:04.740 --> 00:06:07.910
It's just saying that over the
entire interval, the slope

00:06:07.910 --> 00:06:09.180
continues to decrease.

00:06:09.180 --> 00:06:11.610
It goes from very positive,
to less positive, to less

00:06:11.610 --> 00:06:15.010
positive, to zero, to slightly
negative, to more negative, to

00:06:15.010 --> 00:06:18.000
very negative, so it just
decreases the whole time.

00:06:18.000 --> 00:06:21.630
In concave upwards, it
increases the whole time.

00:06:21.630 --> 00:06:23.330
So let me draw an example
that might have the

00:06:23.330 --> 00:06:25.900
combination of the two.

00:06:25.900 --> 00:06:27.510
Maybe I'll leave this
around just so we can

00:06:27.510 --> 00:06:29.680
look at it for reference.

00:06:29.680 --> 00:06:33.160
Let me draw an
axis right there.

00:06:33.160 --> 00:06:36.660
Not the straightest line
in the world, but it'll

00:06:36.660 --> 00:06:38.820
suit our purposes.

00:06:38.820 --> 00:06:41.380
Let me just draw
some curve here.

00:06:41.380 --> 00:06:45.780
So maybe I have something
that looks like that.

00:06:45.780 --> 00:06:48.040
Maybe I have a curve that
looks something like that.

00:06:48.040 --> 00:06:51.770
Now, over what interval of this
function's domain, maybe I'll

00:06:51.770 --> 00:06:55.620
call it h of x, is this curve
concave upwards or

00:06:55.620 --> 00:06:56.890
concave downwards?

00:06:56.890 --> 00:06:59.470
Well, you can look at it just
from inspection, just from

00:06:59.470 --> 00:07:00.680
that original definition.

00:07:00.680 --> 00:07:03.500
I said that concave upwards is
when you're like a U, concave

00:07:03.500 --> 00:07:09.330
downwards is when you're
like an upside down U.

00:07:09.330 --> 00:07:13.380
So over here it looks like
a U from roughly this

00:07:13.380 --> 00:07:15.120
point onwards, right?

00:07:15.120 --> 00:07:17.560
We have this U shape.

00:07:17.560 --> 00:07:19.220
We have this U shape and
maybe the curve just

00:07:19.220 --> 00:07:20.000
keeps going on there.

00:07:20.000 --> 00:07:26.440
So, from here onwards, we are
concave-- let me write it over

00:07:26.440 --> 00:07:29.762
here-- from there onwards, we
are-- I don't know want to

00:07:29.762 --> 00:07:34.625
do it in the same color--
we are concave upwards.

00:07:37.160 --> 00:07:42.050
And then on the interval before
that, from here and before

00:07:42.050 --> 00:07:48.270
this point right here, we
are concave downwards.

00:07:48.270 --> 00:07:51.260
You can just recognize that.

00:07:51.260 --> 00:07:53.260
And let's explore what's
actually happening

00:07:53.260 --> 00:07:54.380
with the slope.

00:07:54.380 --> 00:07:57.146
So over here, the tangent line
has a pretty high-- let me do

00:07:57.146 --> 00:08:00.460
it in a different color-- over
here, the tangent line has a

00:08:00.460 --> 00:08:02.690
pretty high positive slope.

00:08:02.690 --> 00:08:04.420
Then it has a less positive.

00:08:04.420 --> 00:08:06.950
Then at this maximum
point, it goes to zero.

00:08:06.950 --> 00:08:11.150
Then it goes slightly
negative, even more negative,

00:08:11.150 --> 00:08:14.440
even more negative.

00:08:14.440 --> 00:08:19.800
So up until this point, the
slope is decreasing, which is

00:08:19.800 --> 00:08:22.700
completely consistent with
what we said about being

00:08:22.700 --> 00:08:24.240
concave downwards.

00:08:24.240 --> 00:08:27.120
But then something interesting
happens at a point

00:08:27.120 --> 00:08:28.100
right about there.

00:08:28.100 --> 00:08:31.580
I'm not being very accurate or
precise right there, but right

00:08:31.580 --> 00:08:33.930
about there, you see that the
slope of the tangent line

00:08:33.930 --> 00:08:35.680
looks like that, right?

00:08:35.680 --> 00:08:38.560
And that was more negative than
right here, which is more

00:08:38.560 --> 00:08:39.950
negative than right there.

00:08:39.950 --> 00:08:41.770
But over here, something
interesting happens.

00:08:41.770 --> 00:08:45.910
All of a sudden, my slope
starts to increase again.

00:08:45.910 --> 00:08:47.570
It starts to-- I
want to draw it.

00:08:47.570 --> 00:08:50.990
Let me draw it as neatly as
I could possibly draw it.

00:08:50.990 --> 00:08:53.500
So over here, the slope
look something like this.

00:08:58.100 --> 00:09:01.330
So if we go backwards from
there, if we go into more

00:09:01.330 --> 00:09:04.870
negative x's, the x's before
that had a less negative slope,

00:09:04.870 --> 00:09:06.780
so the slope was decreasing.

00:09:06.780 --> 00:09:08.470
But all of a sudden here,
when we go here, the

00:09:08.470 --> 00:09:10.120
slope is slightly higher.

00:09:10.120 --> 00:09:12.630
It's less negative there, then
it's even less negative there,

00:09:12.630 --> 00:09:15.450
then it goes to zero here, then
it goes slightly positive,

00:09:15.450 --> 00:09:17.350
and even more positive.

00:09:17.350 --> 00:09:21.830
So up until this point in
are functions domain, our

00:09:21.830 --> 00:09:23.590
slope was decreasing.

00:09:23.590 --> 00:09:26.750
Our slope was decreasing
up until this point.

00:09:26.750 --> 00:09:29.976
And then after that point,
our slope is increasing.

00:09:32.540 --> 00:09:35.920
So this point, it seems like
we should call it something,

00:09:35.920 --> 00:09:37.650
and we do call it something.

00:09:37.650 --> 00:09:41.320
This point right here, where we
go from concave downwards to

00:09:41.320 --> 00:09:43.440
concave upwards, and it would
actually be true if it was the

00:09:43.440 --> 00:09:47.420
other way around, this is
called an inflection point.

00:09:53.530 --> 00:09:55.520
And this would also be called
an inflection point if

00:09:55.520 --> 00:09:56.240
it was the other way.

00:09:56.240 --> 00:09:58.550
If we went from a curve-- let
me draw a curve that looks like

00:09:58.550 --> 00:10:01.060
this-- that was concave upwards
and then it's concave

00:10:01.060 --> 00:10:05.010
downwards, the point that we're
switching, the point at which

00:10:05.010 --> 00:10:09.260
you go from one to the other,
essentially your rate of the

00:10:09.260 --> 00:10:12.420
change of slope switches signs,
that is an inflection.

00:10:12.420 --> 00:10:14.820
So this is also an
inflection point.

00:10:14.820 --> 00:10:15.660
So what does this mean?

00:10:15.660 --> 00:10:19.480
Hopefully, you understand
visually what concave upwards

00:10:19.480 --> 00:10:21.800
and concave downwards means,
and an inflection point, but

00:10:21.800 --> 00:10:24.520
what does this mean for
the second derivative?

00:10:24.520 --> 00:10:28.500
Remember, the derivative is
the rate of change-- so

00:10:28.500 --> 00:10:31.140
let me write this down.

00:10:31.140 --> 00:10:33.700
So f of x, that's just
the function, right?

00:10:33.700 --> 00:10:35.310
That's just the function.

00:10:35.310 --> 00:10:36.990
That's what we've
been graphing.

00:10:36.990 --> 00:10:42.020
f prime of x, that's the slope
of the function, slope of

00:10:42.020 --> 00:10:45.310
function at any point x.

00:10:45.310 --> 00:10:47.560
You've seen that
multiple times already.

00:10:47.560 --> 00:10:49.900
What's the second derivative?

00:10:49.900 --> 00:10:51.960
Well, you could view it
as the derivative of the

00:10:51.960 --> 00:10:53.430
first derivative, right?

00:10:53.430 --> 00:10:56.440
That's the slope of the
slope, or the rate of

00:10:56.440 --> 00:11:05.440
change of the slope itself.

00:11:05.440 --> 00:11:08.620
So what did we say
about concave upwards?

00:11:08.620 --> 00:11:12.370
concave upwards is when the
slope is increasing, which

00:11:12.370 --> 00:11:15.890
means that the rate of change
of the slope is positive.

00:11:15.890 --> 00:11:19.170
Well, what that means is that
the second derivative, which is

00:11:19.170 --> 00:11:23.010
the slope of the slope, f prime
of x, is going to be

00:11:23.010 --> 00:11:25.090
greater than zero.

00:11:25.090 --> 00:11:28.190
That's what concave upwards
just tells us, right?

00:11:28.190 --> 00:11:31.110
Because if this is greater than
zero, then that means that the

00:11:31.110 --> 00:11:33.420
rate of change of the
slope is positive.

00:11:33.420 --> 00:11:36.540
That's what we said
was concave upwards.

00:11:36.540 --> 00:11:40.590
Now, in concave downwards,
the slope is decreasing.

00:11:40.590 --> 00:11:43.300
The rate of change of the
slope is decreasing,

00:11:43.300 --> 00:11:44.810
or it's a negative.

00:11:44.810 --> 00:11:49.030
So in that case, the
second derivative would

00:11:49.030 --> 00:11:52.020
be less than zero.

00:11:52.020 --> 00:11:55.830
And then in the situation when
we're switching, if over here--

00:11:55.830 --> 00:11:56.830
let's look at this situation.

00:11:56.830 --> 00:11:58.940
We're concave downwards.

00:11:58.940 --> 00:12:01.380
We're concave downwards over
this interval right here.

00:12:01.380 --> 00:12:04.980
That means that the second
derivative has to be less than

00:12:04.980 --> 00:12:06.940
zero over this interval.

00:12:06.940 --> 00:12:10.220
The rate of change of
the slope is negative.

00:12:10.220 --> 00:12:11.760
It's decreasing.

00:12:11.760 --> 00:12:15.050
So over this interval, our
second derivative is less than

00:12:15.050 --> 00:12:17.750
zero, but then we go concave
upwards, so our second

00:12:17.750 --> 00:12:21.510
derivative has to be greater
than zero over this interval,

00:12:21.510 --> 00:12:24.140
which implies if our second
derivative is continuous, that

00:12:24.140 --> 00:12:29.310
the second derivative had to be
equal to zero at that point.

00:12:29.310 --> 00:12:31.710
I'll do that with some
examples in future videos.

00:12:31.710 --> 00:12:35.420
Now, this condition by itself
isn't enough to say that

00:12:35.420 --> 00:12:36.600
it's an inflection point.

00:12:36.600 --> 00:12:39.750
In order for something to be an
inflection point, the second

00:12:39.750 --> 00:12:41.850
derivative has to switch signs.

00:12:41.850 --> 00:12:45.160
It has to go from being
negative in a concave downward

00:12:45.160 --> 00:12:49.670
to positive in concave upwards,
or it has to go from positive

00:12:49.670 --> 00:12:53.100
in concave upwards to negative
in concave downwards.

00:12:53.100 --> 00:12:55.200
And obviously, in between,
it's going to switch.

00:12:55.200 --> 00:12:57.610
If you're switching signs,
you're going to hit zero.

00:12:57.610 --> 00:13:00.860
So when the second derivative
is zero, it might be an

00:13:00.860 --> 00:13:02.940
inflection point, and then you
want to test around it to see

00:13:02.940 --> 00:13:05.280
if it's actually switching
signs, and I'll show you

00:13:05.280 --> 00:13:06.470
that in the next video.

00:13:06.470 --> 00:13:09.240
But hopefully, you at least
have an intuitive sense of what

00:13:09.240 --> 00:13:12.770
inflection points look like and
what the second derivative

00:13:12.770 --> 00:13:13.650
is telling us.

00:13:13.650 --> 00:13:17.160
If the slope is constantly
increasing, then the rate

00:13:17.160 --> 00:13:19.770
of change of slope is
positive, then the second

00:13:19.770 --> 00:13:21.250
derivative is positive.

00:13:21.250 --> 00:13:26.350
If the slope is decreasing,
then the rate of change of

00:13:26.350 --> 00:13:28.760
slope is negative, which
tells us that the second

00:13:28.760 --> 00:13:30.550
derivative is negative.

00:13:30.550 --> 00:13:35.850
And the point in which you're
switching from a positive

00:13:35.850 --> 00:13:37.920
second derivative to a negative
one or a negative one to a

00:13:37.920 --> 00:13:42.140
positive one, that is what is
called an inflection point.