Optimization: cost of materials

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:03.420
A rectangular storage
container with an open top

00:00:03.420 --> 00:00:06.790
needs to have a volume
of 10 cubic meters.

00:00:06.790 --> 00:00:11.290
The length of its base
is twice the width.

00:00:11.290 --> 00:00:14.840
Material for the base
costs $10 per square meter.

00:00:14.840 --> 00:00:17.950
Material for the sides
costs $6 per square meter.

00:00:17.950 --> 00:00:21.440
Find the cost of the material
for the cheapest container.

00:00:21.440 --> 00:00:24.570
So let's draw this
open storage container,

00:00:24.570 --> 00:00:27.040
this open rectangular
storage container.

00:00:27.040 --> 00:00:28.720
So it's going to
have an open top.

00:00:28.720 --> 00:00:33.190
So let me draw its open
top as good as I can.

00:00:33.190 --> 00:00:35.250
So it's going to
have an open top.

00:00:35.250 --> 00:00:37.310
That's the top of my container.

00:00:37.310 --> 00:00:38.945
And then let me draw the sides.

00:00:43.130 --> 00:00:45.330
Just like that.

00:00:45.330 --> 00:00:47.780
So it might look
something like that.

00:00:47.780 --> 00:00:50.960
And then I could draw--
and since it's open top,

00:00:50.960 --> 00:00:53.890
I can see through, I could see
the inside of the container

00:00:53.890 --> 00:00:54.410
as well.

00:00:54.410 --> 00:00:59.774
So the container would
look something like that.

00:00:59.774 --> 00:01:00.940
And so what do they tell us?

00:01:00.940 --> 00:01:03.935
They tell us that the volume
needs to be 10 cubic meters.

00:01:03.935 --> 00:01:05.110
So let me write that down.

00:01:05.110 --> 00:01:09.360
The volume needs to be
equal to 10 meters cubed.

00:01:09.360 --> 00:01:13.040
The length of its base
is twice the width.

00:01:13.040 --> 00:01:17.090
So the length, let's
call the width x,

00:01:17.090 --> 00:01:18.910
so the length is going
to be twice that.

00:01:18.910 --> 00:01:20.350
It's going to be 2x.

00:01:20.350 --> 00:01:22.320
That's what they tell
us right over here.

00:01:22.320 --> 00:01:24.110
They tell us the
material for the base

00:01:24.110 --> 00:01:28.490
costs $10 per square meter.

00:01:28.490 --> 00:01:32.530
So this area right
over here-- [INAUDIBLE]

00:01:32.530 --> 00:01:35.560
if I was transparent I could
continue to draw it down here.

00:01:35.560 --> 00:01:38.010
But this right over
here, that material

00:01:38.010 --> 00:01:40.150
costs $10 per square meter.

00:01:40.150 --> 00:01:45.324
Let me label that
$10 per square meter.

00:01:45.324 --> 00:01:46.990
And then they say
material for the sides

00:01:46.990 --> 00:01:49.830
costs $6 per square meter.

00:01:49.830 --> 00:01:57.680
So the material over here
costs $6 per meter squared.

00:01:57.680 --> 00:02:03.310
So let's see if we can
come up with a value

00:02:03.310 --> 00:02:08.550
or how much this box would cost
to make as a function of x.

00:02:08.550 --> 00:02:10.661
But x only gives us the
dimensions of the base.

00:02:10.661 --> 00:02:12.160
We also need a
dimension for height.

00:02:12.160 --> 00:02:15.320
So it'll be a function
of x and height for now.

00:02:15.320 --> 00:02:18.790
So let's write h as the
height right over here.

00:02:18.790 --> 00:02:21.410
So what is the cost of
this container going to be?

00:02:21.410 --> 00:02:27.090
So the cost is going to be
equal to the cost of the base.

00:02:27.090 --> 00:02:31.750
Well, the cost of the base
is going to be $10 times--

00:02:31.750 --> 00:02:33.290
I'll just write 10.

00:02:33.290 --> 00:02:36.634
This is going to be 10
times the area of the base.

00:02:36.634 --> 00:02:38.050
Well, what's the
area of the base?

00:02:38.050 --> 00:02:40.133
Well, it's going to be the
width times the length.

00:02:40.133 --> 00:02:44.380
So 10 times x times 2x.

00:02:44.380 --> 00:02:46.285
That is the cost of base.

00:02:50.400 --> 00:02:52.872
And now what's going to
be the cost of the sides?

00:02:52.872 --> 00:02:54.330
Well, the different
sides are going

00:02:54.330 --> 00:02:55.690
to have different dimensions.

00:02:55.690 --> 00:03:00.040
You have this side
over here and this side

00:03:00.040 --> 00:03:03.750
over here, which have
the same dimension.

00:03:03.750 --> 00:03:06.370
They both have an
area of x times h.

00:03:06.370 --> 00:03:09.110
You have x times h.

00:03:09.110 --> 00:03:12.490
And then our material
is $6 per square meter.

00:03:12.490 --> 00:03:16.670
So it's 6 times x times h
would be the cost of one

00:03:16.670 --> 00:03:18.540
of these side panels.

00:03:18.540 --> 00:03:20.920
So for two of them we
have to multiply by 2.

00:03:20.920 --> 00:03:24.080
So plus 2 times 6 times h.

00:03:24.080 --> 00:03:25.890
And then we have
these two side panels.

00:03:25.890 --> 00:03:29.070
We have this side
panel right over here

00:03:29.070 --> 00:03:32.340
and we have this side
panel right over here.

00:03:32.340 --> 00:03:36.760
The area of each of them
is going to be 2x times h.

00:03:36.760 --> 00:03:41.380
So it's going to be 2x times h.

00:03:41.380 --> 00:03:43.970
The cost of the material
is going to be 6.

00:03:43.970 --> 00:03:45.430
So the cost of one
of the panels is

00:03:45.430 --> 00:03:49.320
going to be $6 per square
meters times 2xh meters squared.

00:03:49.320 --> 00:03:50.760
But we have two of these panels.

00:03:50.760 --> 00:03:53.060
One panel and two panels.

00:03:53.060 --> 00:03:55.390
So we have to multiply by 2.

00:03:55.390 --> 00:03:58.120
And so we will get-- so
this is right over here,

00:03:58.120 --> 00:03:59.410
this is the cost of the sides.

00:04:03.652 --> 00:04:05.360
And so let's see if
we can simplify this.

00:04:05.360 --> 00:04:06.640
And I'll write it all
in a neutral color.

00:04:06.640 --> 00:04:08.360
So this is going
to be equal to 10.

00:04:08.360 --> 00:04:08.860
Let's see.

00:04:08.860 --> 00:04:10.690
10 times 2 is 20.

00:04:10.690 --> 00:04:13.660
x times x is x squared.

00:04:13.660 --> 00:04:16.620
And then you have
2 times 6 times xh.

00:04:16.620 --> 00:04:22.206
So this is going
to be plus 12xh.

00:04:22.206 --> 00:04:24.080
And then this is going
to be 2 times 6, which

00:04:24.080 --> 00:04:30.560
is 12 times 2 is 24xh plus 24xh.

00:04:30.560 --> 00:04:39.730
So this is going to be equal
to 20x squared plus 36xh.

00:04:39.730 --> 00:04:41.339
So this is going to be my cost.

00:04:41.339 --> 00:04:42.880
But I'm not ready
to optimize it yet.

00:04:42.880 --> 00:04:45.000
We don't know how to optimize
with respect to two variables.

00:04:45.000 --> 00:04:47.416
We only know how to optimize
with respect to one variable,

00:04:47.416 --> 00:04:50.460
and maybe I'll say let's
optimize with respect to x.

00:04:50.460 --> 00:04:52.730
But if we want to optimize
with respect to x,

00:04:52.730 --> 00:04:56.360
we have to express h
as a function of x.

00:04:56.360 --> 00:04:57.650
So how can we do that?

00:04:57.650 --> 00:05:01.030
How can we express h
as a function of x?

00:05:01.030 --> 00:05:05.310
Well, we know that the volume
has to be 10 cubic meters.

00:05:05.310 --> 00:05:13.300
So we know that x, the width
times the length times 2x times

00:05:13.300 --> 00:05:17.010
the height times h
needs to be equal to 10.

00:05:17.010 --> 00:05:19.880
Or another way of saying
that, this tells us

00:05:19.880 --> 00:05:24.210
that 2x squared h,
2x squared times h

00:05:24.210 --> 00:05:25.700
needs to be equal to 10.

00:05:25.700 --> 00:05:27.430
And so if we want h
as a function of x,

00:05:27.430 --> 00:05:30.290
we just divide both
sides by 2x squared.

00:05:30.290 --> 00:05:37.720
And we get h is equal
to 10 over 2x squared.

00:05:37.720 --> 00:05:44.550
Or we could say that h is
equal to 5 over x squared.

00:05:44.550 --> 00:05:47.430
And then we can substitute
back right over here.

00:05:47.430 --> 00:05:50.930
h is equal to 5 over x squared.

00:05:50.930 --> 00:05:55.290
So all of this business is
going to be equal to 20 times

00:05:55.290 --> 00:06:00.950
x squared plus 36 times
x times 5 over x squared.

00:06:05.490 --> 00:06:12.810
So our cost as a function
of x is going to be 20x

00:06:12.810 --> 00:06:15.000
squared 36 times 5.

00:06:15.000 --> 00:06:17.130
Let's see, 30
times 5 is 150 plus

00:06:17.130 --> 00:06:19.340
another 30 is going to be 180.

00:06:19.340 --> 00:06:22.170
So it's going to be plus
180 times, let's see,

00:06:22.170 --> 00:06:27.730
x times x to the negative
2, 180x to the negative x

00:06:27.730 --> 00:06:30.150
to the negative 1 power.

00:06:30.150 --> 00:06:32.340
So we finally have cost
as a function of x.

00:06:32.340 --> 00:06:34.700
Now we're ready to optimize.

00:06:34.700 --> 00:06:36.650
To optimize, we just
have to figure out

00:06:36.650 --> 00:06:38.784
what are the
critical points here

00:06:38.784 --> 00:06:40.325
and whether those
critical points are

00:06:40.325 --> 00:06:42.490
a minimum or a maximum value.

00:06:42.490 --> 00:06:44.972
So let's see what we can do.

00:06:44.972 --> 00:06:47.430
So to find a critical point,
we take the derivative, figure

00:06:47.430 --> 00:06:50.357
out where the derivative
is undefined or equal to 0,

00:06:50.357 --> 00:06:52.190
and those are our
candidate critical points.

00:06:52.190 --> 00:06:54.060
And then from the
critical points

00:06:54.060 --> 00:06:57.040
we find, they might be
minimum or maximum values.

00:06:57.040 --> 00:07:00.980
So the derivative of c of
our cost with respect to x

00:07:00.980 --> 00:07:09.310
is going to be equal to 40
times x minus 180 times x

00:07:09.310 --> 00:07:11.450
to the negative 2 power.

00:07:11.450 --> 00:07:16.730
Now, this seems-- well
it's defined for all x

00:07:16.730 --> 00:07:18.910
except for x equaling 0.

00:07:18.910 --> 00:07:21.157
But x equaling 0 is
not interesting to us

00:07:21.157 --> 00:07:22.740
as a critical point
because then we're

00:07:22.740 --> 00:07:23.864
going to have a degenerate.

00:07:23.864 --> 00:07:25.454
This is going to
have no base at all.

00:07:25.454 --> 00:07:27.620
So we don't want to worry
about that critical point.

00:07:27.620 --> 00:07:30.009
We would have no volume at
all, so it would not work out.

00:07:30.009 --> 00:07:31.800
And actually, if x
equals 0 then our height

00:07:31.800 --> 00:07:34.260
is undefined as well.

00:07:34.260 --> 00:07:36.780
So this was defined
for everything else,

00:07:36.780 --> 00:07:39.410
for anything other
than x equals 0.

00:07:39.410 --> 00:07:41.520
So let's see when
this derivative is

00:07:41.520 --> 00:07:46.590
equal to 0 in our search for
our potential critical points.

00:07:46.590 --> 00:07:48.900
So when does-- I'll
do it over here.

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When does 40x minus 180x
to the negative 2 equal 0?

00:07:57.760 --> 00:08:00.530
Well, we could add the 180x to
the negative 2 to both sides.

00:08:00.530 --> 00:08:04.310
We get 40x is equal to 180.

00:08:04.310 --> 00:08:10.010
And I could write it
as 180 over x squared.

00:08:10.010 --> 00:08:10.740
Now let's see.

00:08:10.740 --> 00:08:14.940
We could multiply both sides
of this equation by x squared

00:08:14.940 --> 00:08:21.430
and we would get 40x to
the third is equal to 180.

00:08:21.430 --> 00:08:23.680
Divide both sides by 40.

00:08:23.680 --> 00:08:29.040
You get x to the third
is equal to 180 over 40,

00:08:29.040 --> 00:08:32.400
which is the same
thing as 18 over 4,

00:08:32.400 --> 00:08:35.350
which is the same
thing as 9 over 2.

00:08:35.350 --> 00:08:36.980
And so if we want
to solve for x,

00:08:36.980 --> 00:08:41.730
we get that x is equal
to a critical point.

00:08:41.730 --> 00:08:47.370
We get a critical point of x is
equal to 9/2 to the 1/3 power,

00:08:47.370 --> 00:08:48.689
the cube root of 9/2.

00:08:48.689 --> 00:08:49.230
So let's see.

00:08:49.230 --> 00:08:52.600
Let's get an approximate
value for what that is.

00:08:52.600 --> 00:08:58.150
So if we take 9/2,
9 divided by 2--

00:08:58.150 --> 00:08:59.820
I guess you could
call that 4.5--

00:08:59.820 --> 00:09:02.670
and we want to raise
it to the 1/3 power.

00:09:02.670 --> 00:09:06.860
To the 1/3 power we get 1.65.

00:09:06.860 --> 00:09:12.925
So it's approximately equal
to 1.65 as our critical point.

00:09:12.925 --> 00:09:14.300
Now the way the
problem is asked,

00:09:14.300 --> 00:09:17.280
we're only getting one
legitimate critical point here.

00:09:17.280 --> 00:09:19.840
So that's probably going
to be the x at which we

00:09:19.840 --> 00:09:23.000
achieve a minimum value.

00:09:23.000 --> 00:09:25.930
But let's use our second
derivative test just in case

00:09:25.930 --> 00:09:27.500
to make sure that
we're definitely

00:09:27.500 --> 00:09:29.830
concave upwards over here,
in which case, this will

00:09:29.830 --> 00:09:32.160
definitely be the
x value at which we

00:09:32.160 --> 00:09:33.960
achieve a minimum value.

00:09:33.960 --> 00:09:35.615
So the second derivative.

00:09:35.615 --> 00:09:36.740
I'll do it right over here.

00:09:36.740 --> 00:09:39.060
The second derivative
of our cost function

00:09:39.060 --> 00:09:40.850
is just the derivative
of this, which

00:09:40.850 --> 00:09:49.770
is going to be equal to 40 minus
180 times negative 2, which

00:09:49.770 --> 00:09:51.660
is negative 360.

00:09:51.660 --> 00:09:58.960
So it's going to be plus
360 over x to the 3.

00:09:58.960 --> 00:10:01.970
The derivative of this is
negative 2 times negative 180,

00:10:01.970 --> 00:10:04.460
which is positive 360x to
the negative 3 power, which

00:10:04.460 --> 00:10:06.630
is exactly this right over here.

00:10:06.630 --> 00:10:10.200
So when x is equal to 1.65,
this is going to be positive.

00:10:10.200 --> 00:10:11.740
This is going to be positive.

00:10:11.740 --> 00:10:15.620
So let me write this down.
c prime prime of 1.65

00:10:15.620 --> 00:10:18.050
is definitely greater than 0.

00:10:18.050 --> 00:10:22.320
So we're definitely concave
upwards when x is 1.65.

00:10:22.320 --> 00:10:25.500
Concave upwards, which
means that our graph is

00:10:25.500 --> 00:10:27.260
going to look
something like this.

00:10:27.260 --> 00:10:28.680
And so where the
derivative equal

00:10:28.680 --> 00:10:32.540
to 0, which is right over there,
we are at a minimum point.

00:10:32.540 --> 00:10:35.424
We are minimizing our cost.

00:10:35.424 --> 00:10:37.590
And so if we go back to the
question, the only thing

00:10:37.590 --> 00:10:39.750
that we have to do now--
We know the x value

00:10:39.750 --> 00:10:41.040
that minimizes our cost.

00:10:41.040 --> 00:10:44.070
We now have to find the cost of
the material for the cheapest

00:10:44.070 --> 00:10:44.920
container.

00:10:44.920 --> 00:10:48.456
So we just have to figure
out what our cost is.

00:10:48.456 --> 00:10:49.830
And we already
know what our cost

00:10:49.830 --> 00:10:51.460
is as a function
of x, so we just

00:10:51.460 --> 00:10:55.240
have to put 1.65
into this equation.

00:10:55.240 --> 00:10:57.270
Evaluate the function at 1.65.

00:10:57.270 --> 00:10:59.690
So let's do that.

00:10:59.690 --> 00:11:05.520
Our cost is going to be
equal to 20 times 1.65.

00:11:05.520 --> 00:11:07.020
I should say
approximately equal to,

00:11:07.020 --> 00:11:08.436
because I'm using
an approximation

00:11:08.436 --> 00:11:09.860
of this original value.

00:11:09.860 --> 00:11:18.870
1.65 squared plus 180.

00:11:18.870 --> 00:11:20.681
I could say divided by 1.65.

00:11:20.681 --> 00:11:22.180
That's the same
thing as multiplying

00:11:22.180 --> 00:11:24.140
by 1.65 to the negative 1.

00:11:24.140 --> 00:11:33.520
So divided by 1.65,
which is equal to 163.

00:11:33.520 --> 00:11:37.370
I'll just say $163.5.

00:11:37.370 --> 00:11:38.670
So it's approximately.

00:11:38.670 --> 00:11:42.100
So the cost-- let me
do this in a new color.

00:11:42.100 --> 00:11:44.420
We deserve a drum roll now.

00:11:44.420 --> 00:11:57.770
The cost when x is 1.65 is
approximately equal to $163.54.

00:11:57.770 --> 00:12:05.300
So $163.54, which is
quite an expensive box.

00:12:05.300 --> 00:12:07.610
So this is kind of expensive.

00:12:07.610 --> 00:12:10.370
This is fairly
expensive material here.

00:12:10.370 --> 00:12:11.820
Although it's a
fairly large box.

00:12:11.820 --> 00:12:16.410
1.65 meters in
width, and it's going

00:12:16.410 --> 00:12:18.110
to be twice that in length.

00:12:18.110 --> 00:12:20.830
And then you could figure out
what its height is going to be.

00:12:20.830 --> 00:12:22.620
Although it's not
going to be too tall.

00:12:22.620 --> 00:12:24.567
5 divided by 1.65 squared.

00:12:24.567 --> 00:12:27.150
I don't know, it'll be roughly
a little under two meters tall.

00:12:27.150 --> 00:12:30.485
So it actually is quite
a large box made out

00:12:30.485 --> 00:12:32.310
of quite expensive material.

00:12:32.310 --> 00:12:37.240
The minimum cost to make this
box is going to be $163.54.


Optimization: profit

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:02.700
You've opened up a
shoe factory and you're

00:00:02.700 --> 00:00:06.020
trying to figure out how many
thousands of pairs of shoes

00:00:06.020 --> 00:00:08.980
to produce in order to
optimize your profit.

00:00:08.980 --> 00:00:13.505
And so let's let x equal the
thousands of pairs produced.

00:00:20.141 --> 00:00:21.640
Now let's think
about how much money

00:00:21.640 --> 00:00:22.990
you're going to make per pair.

00:00:22.990 --> 00:00:24.573
Actually, let me say
how much revenue,

00:00:24.573 --> 00:00:28.079
which is how much you actually
get to sell those shoes for.

00:00:28.079 --> 00:00:29.620
So let's write a
function right here.

00:00:29.620 --> 00:00:31.910
Revenue as a function of x.

00:00:31.910 --> 00:00:33.470
Well, you have a
wholesaler who's

00:00:33.470 --> 00:00:36.720
willing to pay you $10
per pair for as many pairs

00:00:36.720 --> 00:00:38.090
as you're willing to give him.

00:00:38.090 --> 00:00:42.100
So your revenue as a function
of x is going to be 10 times x.

00:00:42.100 --> 00:00:45.490
And since x is in thousands
of pairs produced, if x is 1,

00:00:45.490 --> 00:00:50.500
that means 1,000 pairs produced
times 10, which means $10,000.

00:00:50.500 --> 00:00:51.980
But this will just give you 10.

00:00:51.980 --> 00:00:54.350
So this right over here is
in thousands of dollars.

00:01:00.060 --> 00:01:02.570
So if x is 1, that means
1,000 pairs produced.

00:01:02.570 --> 00:01:05.260
10 times 1 says
r is equal to 10,

00:01:05.260 --> 00:01:09.689
but that really means $10,000.

00:01:09.689 --> 00:01:11.480
Now, it would be a nice
business if all you

00:01:11.480 --> 00:01:13.210
had was revenue and no costs.

00:01:13.210 --> 00:01:14.465
But you do have costs.

00:01:14.465 --> 00:01:16.590
You have materials, you
have to build your factory,

00:01:16.590 --> 00:01:18.089
have to pay your
employees, you have

00:01:18.089 --> 00:01:19.500
to pay the electricity bill.

00:01:19.500 --> 00:01:21.210
And so you hire a
bunch of consultants

00:01:21.210 --> 00:01:24.945
to come up with what your
cost is as a function of x.

00:01:28.170 --> 00:01:29.600
And they come up
with a function.

00:01:29.600 --> 00:01:33.270
They say it is the number of the
thousands of pairs you produce

00:01:33.270 --> 00:01:36.920
cubed minus 6 times the
thousands of pairs you produce

00:01:36.920 --> 00:01:41.950
squared plus 15 times
the thousands of pairs

00:01:41.950 --> 00:01:43.300
that you produce.

00:01:43.300 --> 00:01:46.580
And once again,
this is also going

00:01:46.580 --> 00:01:49.710
to be in thousands of dollars.

00:01:49.710 --> 00:01:54.682
Now, given these functions
of x for revenue and cost,

00:01:54.682 --> 00:01:58.930
what is profit as a
function of x going to be?

00:01:58.930 --> 00:02:03.580
Well, your profit
as a function of x

00:02:03.580 --> 00:02:07.710
is just going to be equal to
your revenue as a function of x

00:02:07.710 --> 00:02:09.800
minus your cost as
a function of x.

00:02:12.592 --> 00:02:14.300
If you produce a
certain amount and let's

00:02:14.300 --> 00:02:17.330
say you bring in, I don't
know, $10,000 of revenue

00:02:17.330 --> 00:02:19.970
and it costs you $5,000
to produce those shoes,

00:02:19.970 --> 00:02:21.529
you'll have $5,000 in profit.

00:02:21.529 --> 00:02:23.570
Those numbers aren't the
ones that would actually

00:02:23.570 --> 00:02:25.250
you would get from
this right here.

00:02:25.250 --> 00:02:27.230
I'm just giving you an example.

00:02:27.230 --> 00:02:28.990
So this is what you
want to optimize.

00:02:28.990 --> 00:02:36.170
You want to optimize
p as a function of x.

00:02:36.170 --> 00:02:37.140
So what is it?

00:02:37.140 --> 00:02:39.070
I've just said it here
in abstract terms,

00:02:39.070 --> 00:02:41.170
but we know what r of
x is and what's c of x.

00:02:41.170 --> 00:02:46.380
This is 10x minus
all of this business.

00:02:46.380 --> 00:02:53.590
So minus x to the third
plus 6x squared minus 15x.

00:02:53.590 --> 00:02:56.110
I just subtracted x
squared, you subtract

00:02:56.110 --> 00:02:58.880
6x squared it becomes positive,
you subtract a 15x it becomes

00:02:58.880 --> 00:03:03.520
negative 15x, and then we can
simplify this as-- let's see,

00:03:03.520 --> 00:03:07.200
we have negative x
to the third plus 6x

00:03:07.200 --> 00:03:12.370
squared minus 15x plus
10x, so that is minus 5x.

00:03:12.370 --> 00:03:16.497
Now if we want to optimize this
profit function analytically,

00:03:16.497 --> 00:03:18.080
the easiest way is
to think about what

00:03:18.080 --> 00:03:20.240
are the critical points
of this profit function

00:03:20.240 --> 00:03:21.740
and are any of those
critical points

00:03:21.740 --> 00:03:23.430
minimum points or
maximum points?

00:03:23.430 --> 00:03:26.870
And if one of them is a
maximum point, then we can say,

00:03:26.870 --> 00:03:28.190
well, let's produce that many.

00:03:28.190 --> 00:03:30.292
That is going to be--
we will have optimized

00:03:30.292 --> 00:03:32.500
or we will figure out the
quantity we need to produce

00:03:32.500 --> 00:03:35.020
in order to optimize our profit.

00:03:35.020 --> 00:03:36.710
So to figure out
critical points,

00:03:36.710 --> 00:03:39.600
we essentially have to find
the derivative of our function

00:03:39.600 --> 00:03:42.110
and figure out when does
that derivative equal 0

00:03:42.110 --> 00:03:44.070
or when is that
derivative undefined?

00:03:44.070 --> 00:03:46.490
That's the definition
of critical points.

00:03:46.490 --> 00:03:50.960
So p prime of x is going
to be equal to negative 3x

00:03:50.960 --> 00:03:55.790
squared plus 12x minus 5.

00:03:55.790 --> 00:03:58.100
And so this thing is going
to be defined for all x.

00:03:58.100 --> 00:04:00.140
So the only critical
points we're going to have

00:04:00.140 --> 00:04:04.280
is when the first derivative
right over here is equal to 0.

00:04:04.280 --> 00:04:10.470
So negative 3x squared plus 12x
minus 5 needs to be equal to 0

00:04:10.470 --> 00:04:12.190
in order for x to
be a critical point.

00:04:12.190 --> 00:04:14.300
So now we just have
to solve for x.

00:04:14.300 --> 00:04:17.829
And so we just are essentially
solving a quadratic equation.

00:04:17.829 --> 00:04:19.860
Just so that I don't
have as many negatives,

00:04:19.860 --> 00:04:21.820
let's multiply both
sides by negative 1.

00:04:21.820 --> 00:04:24.040
I just like to have a
clean first coefficient.

00:04:24.040 --> 00:04:25.950
So if we multiply both
sides by negative 1,

00:04:25.950 --> 00:04:33.330
we get 3x squared minus
12x plus 5 is equal to 0.

00:04:33.330 --> 00:04:36.700
And now we can use the quadratic
formula to solve for x.

00:04:36.700 --> 00:04:41.960
So x is going to be equal
to negative b, which is 12,

00:04:41.960 --> 00:04:43.910
plus or minus the square root.

00:04:43.910 --> 00:04:46.650
I always need to make my
radical signs wide enough.

00:04:46.650 --> 00:04:48.840
The square root of
b squared, which

00:04:48.840 --> 00:04:56.340
is 144, minus 4 times a, which
is 3, times c, which is 5.

00:04:59.340 --> 00:05:01.020
All of that over 2a.

00:05:01.020 --> 00:05:03.830
So 2 times 3 is 6.

00:05:03.830 --> 00:05:07.250
So x is equal to
12 plus or minus

00:05:07.250 --> 00:05:12.250
the square root of, let's see,
4 times 3 is 12 times 5 is 60.

00:05:12.250 --> 00:05:16.190
144 minus 60 is 84.

00:05:16.190 --> 00:05:17.990
All of that over 6.

00:05:17.990 --> 00:05:23.430
So x could be equal to 12 plus
the square root of 84 over 6

00:05:23.430 --> 00:05:29.930
or x could be equal to 12 minus
the square root of 84 over 6.

00:05:29.930 --> 00:05:31.580
So let's figure out
what these two are.

00:05:31.580 --> 00:05:34.270
And I'll use a calculator.

00:05:34.270 --> 00:05:37.140
I'll use the calculator
for this one.

00:05:37.140 --> 00:05:49.940
So I get, let's see, 12 plus the
square root of 84 divided by 6

00:05:49.940 --> 00:05:54.810
gives me 3.5--
I'll just say 3.53.

00:05:54.810 --> 00:05:59.440
So approximately
3.-- Actually, let me

00:05:59.440 --> 00:06:02.550
go one more digit, because
I'm talking about thousands.

00:06:02.550 --> 00:06:11.730
So let me say 3.528.

00:06:11.730 --> 00:06:15.050
So this would literally
be 3,528 shoes,

00:06:15.050 --> 00:06:19.170
because this is in
thousands, or pairs of shoes.

00:06:19.170 --> 00:06:21.370
And then let's do the
situation where we subtract.

00:06:21.370 --> 00:06:24.520
And actually we can look
at our previous entry

00:06:24.520 --> 00:06:27.220
and just change this
to a subtraction.

00:06:27.220 --> 00:06:30.570
Change that to not a
negative sign, a subtraction.

00:06:30.570 --> 00:06:31.530
There you go.

00:06:31.530 --> 00:06:37.470
And we get 0.4725.

00:06:37.470 --> 00:06:38.410
Let me remember that.

00:06:38.410 --> 00:06:41.810
0.4725.

00:06:41.810 --> 00:06:47.912
Approximately equal to 0.4725.

00:06:47.912 --> 00:06:49.370
I have a horrible
memory, so let me

00:06:49.370 --> 00:06:50.828
review that I wrote
the same thing.

00:06:50.828 --> 00:06:53.860
4725.

00:06:53.860 --> 00:06:54.360
Yep.

00:06:54.360 --> 00:06:55.470
All right.

00:06:55.470 --> 00:06:57.275
Now these are all
we know about these,

00:06:57.275 --> 00:06:58.710
or these are both
critical points.

00:06:58.710 --> 00:07:02.530
These are points at which
our derivative is equal to 0.

00:07:02.530 --> 00:07:04.680
But we don't know whether
they're minimum points,

00:07:04.680 --> 00:07:06.810
they're points at which
the function takes

00:07:06.810 --> 00:07:09.930
on a minimum value, a
maximum value, or neither.

00:07:09.930 --> 00:07:11.980
To do that, I'll use the
second derivative test

00:07:11.980 --> 00:07:14.210
to figure out if our
function is concave upwards

00:07:14.210 --> 00:07:17.750
or concave downwards or
neither at one of these points.

00:07:17.750 --> 00:07:20.630
So let's look at the
second derivative.

00:07:20.630 --> 00:07:24.530
So p prime prime
of x is going to be

00:07:24.530 --> 00:07:32.140
equal to negative 6x plus 12.

00:07:32.140 --> 00:07:37.440
And so if we look at-- let me
make sure I have enough space.

00:07:37.440 --> 00:07:44.660
So if we look at p
prime prime of 3.528.

00:07:44.660 --> 00:07:46.860
So let's see if I can
think this through.

00:07:46.860 --> 00:07:49.430
So this is between 3 and 4.

00:07:49.430 --> 00:07:52.700
So if we take the lower
value, 3 times negative 6

00:07:52.700 --> 00:07:56.180
is negative 18 plus 12 is
going to be less than 0.

00:07:56.180 --> 00:07:58.170
And if this was 4 it'd
be even more negative,

00:07:58.170 --> 00:07:59.878
so this thing is going
to be less than 0.

00:07:59.878 --> 00:08:02.301
Don't even have to use my
calculator to evaluate it.

00:08:02.301 --> 00:08:04.050
Now what about this
thing right over here?

00:08:04.050 --> 00:08:05.690
0.47.

00:08:05.690 --> 00:08:09.890
Well, 0.47, that's roughly 0.5.

00:08:09.890 --> 00:08:12.954
So negative 6 times
0.5 is negative 3.

00:08:12.954 --> 00:08:15.120
This is going to be nowhere
close to being negative.

00:08:15.120 --> 00:08:17.080
This is definitely
going to be positive.

00:08:17.080 --> 00:08:24.280
So p prime prime of
0.4725 is greater than 0.

00:08:24.280 --> 00:08:27.070
So the fact that the second
derivative is less than 0,

00:08:27.070 --> 00:08:31.850
that means that my
derivative is decreasing.

00:08:31.850 --> 00:08:33.840
My first derivative
is decreasing

00:08:33.840 --> 00:08:37.010
when x is equal to
this value, which

00:08:37.010 --> 00:08:39.830
means that our
graph, our function,

00:08:39.830 --> 00:08:41.590
is concave downwards here.

00:08:46.420 --> 00:08:50.650
And concave downwards means
it looks something like this.

00:08:50.650 --> 00:08:52.920
And so you can see what it
looks something like that,

00:08:52.920 --> 00:08:56.277
the slope is
constantly decreasing.

00:08:56.277 --> 00:08:58.610
So if you have an interval
where the slope is decreasing

00:08:58.610 --> 00:09:02.150
and you know the point where
the slope is exactly 0, which

00:09:02.150 --> 00:09:09.860
is where x is equal to
3.528, it must be a maximum.

00:09:09.860 --> 00:09:14.190
So we actually do take on a
maximum value when x is 3.528.

00:09:14.190 --> 00:09:17.124
On the other side we
see that over here we're

00:09:17.124 --> 00:09:17.790
concave upwards.

00:09:22.330 --> 00:09:24.751
The graph will look something
like this over here.

00:09:24.751 --> 00:09:27.000
And if the slope is 0 where
the graph looks like that,

00:09:27.000 --> 00:09:30.110
we see that that
is a local minimum.

00:09:33.199 --> 00:09:34.990
And so we definitely
don't want to do this.

00:09:34.990 --> 00:09:40.400
We would produce
472 and 1/2 units

00:09:40.400 --> 00:09:43.300
if we were looking to minimize
our profit, maximize our loss.

00:09:43.300 --> 00:09:45.672
So we definitely
don't want to do this.

00:09:45.672 --> 00:09:47.130
But let's actually
think about what

00:09:47.130 --> 00:09:49.600
our profit is going
to be if we produce

00:09:49.600 --> 00:09:55.480
3.528 thousands of
shoes, or 3,528 shoes.

00:09:55.480 --> 00:09:57.570
Well, to do that we just
have to input it back

00:09:57.570 --> 00:10:00.450
into our original profit
function right over here.

00:10:00.450 --> 00:10:01.780
So let's do that.

00:10:01.780 --> 00:10:04.160
So I get my calculator out.

00:10:04.160 --> 00:10:08.090
So my original profit
function is right over there.

00:10:08.090 --> 00:10:09.950
So I want to be able
to see that and that.

00:10:09.950 --> 00:10:24.570
So I get negative 3.528 to the
third power plus 6 times 3.528

00:10:24.570 --> 00:10:32.280
squared minus 5 times
3.528 gives me--

00:10:32.280 --> 00:10:39.990
and we get a drum roll now--
gives me a profit of 13.128.

00:10:39.990 --> 00:10:41.460
So let me write this down.

00:10:41.460 --> 00:10:48.570
The profit when I produce 3,528
shoes is approximately equal to

00:10:48.570 --> 00:10:51.860
or it is equal to, if I produce
exactly that many shoes,

00:10:51.860 --> 00:10:56.389
it's equal to 13.128.

00:10:56.389 --> 00:10:58.430
Or actually it's approximately,
because I'm still

00:10:58.430 --> 00:11:02.310
rounding 13.128.

00:11:02.310 --> 00:11:07.980
So if I produce 3,528
shoes in a given period,

00:11:07.980 --> 00:11:12.470
I'm going to have a
profit of $13,128.

00:11:12.470 --> 00:11:14.540
Remember, this right over
here is in thousands,

00:11:14.540 --> 00:11:17.590
this right over here
is 13.128 thousands

00:11:17.590 --> 00:11:21.850
of dollars in profit,
which is $13,128.

00:11:21.850 --> 00:11:25.811
Anyway, we are now going to
be rich shoe manufacturers.


Optimization: box volume (Part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:05.900
Let's say that we have
a sheet of cardboard

00:00:05.900 --> 00:00:08.620
that is 20 inches by 30 inches.

00:00:08.620 --> 00:00:11.535
Let me draw the cardboard
as neatly as I can.

00:00:11.535 --> 00:00:13.810
So it might look
something like that.

00:00:13.810 --> 00:00:16.846
So that is my
sheet of cardboard.

00:00:16.846 --> 00:00:18.720
And just to make sure
we know the dimensions,

00:00:18.720 --> 00:00:22.150
there's 20 inches by 30 inches.

00:00:22.150 --> 00:00:24.240
And what we're
going to do is cut

00:00:24.240 --> 00:00:26.050
out the corners
of this cardboard.

00:00:26.050 --> 00:00:28.960
And all the corners are
going to be squares,

00:00:28.960 --> 00:00:31.870
and we're going to cut
out an x by x corner

00:00:31.870 --> 00:00:39.380
from each of the corners of this
piece of cardboard-- x by x.

00:00:39.380 --> 00:00:46.034
Over here, x by x, and
then over here, x by x.

00:00:46.034 --> 00:00:48.200
And what we'll do is after
we cut out those corners,

00:00:48.200 --> 00:00:51.260
we can essentially
fold down the flaps.

00:00:51.260 --> 00:00:52.240
Let me draw the flap.

00:00:52.240 --> 00:00:54.740
So you could imagine we
can fold right there,

00:00:54.740 --> 00:00:58.210
we could fold right there,
we could fold right there,

00:00:58.210 --> 00:00:59.600
and we would form a box.

00:00:59.600 --> 00:01:02.950
I guess you could imagine a
box without a bottom to it,

00:01:02.950 --> 00:01:05.170
or you can view a box
without a top to it.

00:01:05.170 --> 00:01:07.130
So if we were to
fold everything up,

00:01:07.130 --> 00:01:10.490
we would get a container that
looks something like this.

00:01:10.490 --> 00:01:12.770
Let me make my best
attempt to draw it.

00:01:12.770 --> 00:01:15.340
So it would look like this.

00:01:18.250 --> 00:01:20.500
This is one flap folded up.

00:01:20.500 --> 00:01:22.920
You can imagine this
flap right over here,

00:01:22.920 --> 00:01:25.170
if I were to fold
it up like that,

00:01:25.170 --> 00:01:28.230
it now would look like this.

00:01:28.230 --> 00:01:29.580
It would now look like this.

00:01:29.580 --> 00:01:32.190
The height of the flap is x.

00:01:32.190 --> 00:01:36.180
So this distance
right over here is x.

00:01:36.180 --> 00:01:39.690
And then if I were to fold
this flap, if I were-- let

00:01:39.690 --> 00:01:41.920
me do that a little bit neater.

00:01:41.920 --> 00:01:44.460
If I were to fold this
flap right over here,

00:01:44.460 --> 00:01:47.735
if I were to fold that up,
then it would look like this.

00:01:50.250 --> 00:01:53.420
Let me make my best
attempt to draw it.

00:01:53.420 --> 00:01:57.970
It would look like that.

00:01:57.970 --> 00:02:01.460
And then I would fold
that back flap up.

00:02:01.460 --> 00:02:05.896
So the back flap would
look something like that.

00:02:05.896 --> 00:02:08.690
That would be the back flap.

00:02:08.690 --> 00:02:10.690
It would look
something like that.

00:02:10.690 --> 00:02:12.940
And then this flap
over here would--

00:02:12.940 --> 00:02:19.930
if I fold it up would
look something like that.

00:02:19.930 --> 00:02:22.620
And then of course, my
base of my whole thing,

00:02:22.620 --> 00:02:26.590
so this whole region right over
here of my piece of cardboard,

00:02:26.590 --> 00:02:31.300
would be the floor of this
box that I'm constructing.

00:02:31.300 --> 00:02:33.530
And what I want to do
is I want to maximize

00:02:33.530 --> 00:02:35.240
the volume of this box.

00:02:35.240 --> 00:02:37.840
I want to maximize
how much it can hold.

00:02:37.840 --> 00:02:41.980
And I want to maximize it by
picking my x appropriately.

00:02:41.980 --> 00:02:44.540
So let's think about what
the volume of this box

00:02:44.540 --> 00:02:45.784
is as a function of x.

00:02:45.784 --> 00:02:47.200
Well, in order to
do that, we have

00:02:47.200 --> 00:02:51.620
to figure out all the dimensions
of this box as a function of x.

00:02:51.620 --> 00:02:54.600
We already know that this
corner right over here,

00:02:54.600 --> 00:02:57.510
which is made up of when
this side and this side

00:02:57.510 --> 00:02:59.684
connect when you fold
these two flaps up,

00:02:59.684 --> 00:03:01.600
that's going to be the
same height over there.

00:03:01.600 --> 00:03:03.516
That's going to be the
same height over there.

00:03:03.516 --> 00:03:05.770
The height of this
box is going to be x.

00:03:05.770 --> 00:03:07.490
But what's the width?

00:03:07.490 --> 00:03:11.672
What is the width of
this box going to be?

00:03:11.672 --> 00:03:13.130
Well, the width of
the box is going

00:03:13.130 --> 00:03:16.440
to be this distance
right over here.

00:03:16.440 --> 00:03:21.970
And this distance is going
to be 20 inches minus not 1x,

00:03:21.970 --> 00:03:23.930
but minus 2 x's.

00:03:23.930 --> 00:03:27.285
So this is going
to be 20 minus 2x.

00:03:27.285 --> 00:03:28.410
You see it right over here.

00:03:28.410 --> 00:03:30.030
This whole distance is 20.

00:03:30.030 --> 00:03:31.990
You subtract this x,
you subtract this x,

00:03:31.990 --> 00:03:34.370
and you get this
distance right over here.

00:03:34.370 --> 00:03:36.670
So it's 20 minus 2x.

00:03:36.670 --> 00:03:41.260
Now, the same logic, what
is the depth of the box?

00:03:41.260 --> 00:03:44.990
What is that distance
right over there?

00:03:44.990 --> 00:03:50.030
Well, that distance is this
distance right over here.

00:03:50.030 --> 00:03:53.090
We know that this entire
distance is 30 inches.

00:03:53.090 --> 00:03:56.870
If we subtract out this x
and we subtract out this x,

00:03:56.870 --> 00:03:58.910
we get the distance
that we care about.

00:03:58.910 --> 00:04:04.280
So this is going
to be 30 minus 2x.

00:04:04.280 --> 00:04:06.420
So now we have all
of the dimensions.

00:04:06.420 --> 00:04:09.850
So what would the volume
be as a function of x?

00:04:09.850 --> 00:04:14.120
Well, the volume
as a function of x

00:04:14.120 --> 00:04:16.490
is going to be equal
to the height, which

00:04:16.490 --> 00:04:24.430
is x, times the width, which
is 20 minus x-- sorry, 20

00:04:24.430 --> 00:04:28.230
minus 2x times the depth,
which is 30 minus 2x.

00:04:32.500 --> 00:04:37.460
Now, what are possible values of
x that give us a valid volume?

00:04:37.460 --> 00:04:41.390
Well, x can't be less than 0.

00:04:41.390 --> 00:04:42.970
You can't make a
negative cut here.

00:04:42.970 --> 00:04:45.780
Somehow we would have to add
cardboard or something there.

00:04:45.780 --> 00:04:48.680
So we know that x is going to
be greater than or equal to 0.

00:04:48.680 --> 00:04:50.370
So let me write this down.

00:04:50.370 --> 00:04:54.440
x is going to be greater
than or equal to 0.

00:04:54.440 --> 00:04:56.700
And what does it
have to be less than?

00:04:56.700 --> 00:04:59.270
Well, I can cut at
most-- we can see here

00:04:59.270 --> 00:05:02.270
the length right over here, this
pink color, this mauve color,

00:05:02.270 --> 00:05:04.330
is 20 minus 2x.

00:05:04.330 --> 00:05:06.170
So this has got to
be greater than 0.

00:05:06.170 --> 00:05:09.200
This is always going to be
shorter than the 30 minus 2x,

00:05:09.200 --> 00:05:15.460
but the 20 minus 2x has to be
greater than or equal to 0.

00:05:15.460 --> 00:05:17.870
You can't cut more
cardboard than there is.

00:05:17.870 --> 00:05:21.720
Or you could say that 20 has to
be greater than or equal to 2x,

00:05:21.720 --> 00:05:24.720
or you could say
that 10 is going

00:05:24.720 --> 00:05:26.910
to be greater than
or equal to x, which

00:05:26.910 --> 00:05:30.510
is another way of saying that
x is going to be less than

00:05:30.510 --> 00:05:32.570
or equal to 10.

00:05:32.570 --> 00:05:34.190
That's a different
shade of yellow.

00:05:34.190 --> 00:05:36.950
x is going to be less
than or equal to 10.

00:05:36.950 --> 00:05:39.800
So x has got to be
between 0 and 10.

00:05:39.800 --> 00:05:42.850
Otherwise we've cut too
much, or we're somehow

00:05:42.850 --> 00:05:44.940
adding cardboard or something.

00:05:44.940 --> 00:05:49.430
So first let's think about
the volume at the endpoints

00:05:49.430 --> 00:05:51.860
of our-- essentially of
our domain, of what x

00:05:51.860 --> 00:05:53.860
can be for our volume.

00:05:53.860 --> 00:05:59.670
Well, our volume when x is
equal to 0 is equal to what?

00:05:59.670 --> 00:06:01.331
We can have 0 times
all of this stuff.

00:06:01.331 --> 00:06:02.330
And it's pretty obvious.

00:06:02.330 --> 00:06:04.060
You're not going to
have any height here.

00:06:04.060 --> 00:06:05.684
So you're not going
to have any volume,

00:06:05.684 --> 00:06:07.440
so our volume would be 0.

00:06:07.440 --> 00:06:11.120
What is our volume
when x is equal to 10?

00:06:11.120 --> 00:06:15.060
Well, if x equaled 10,
then the width here

00:06:15.060 --> 00:06:17.820
that I've drawn in
pink would be 0.

00:06:17.820 --> 00:06:19.984
So once again, we
would have no volume.

00:06:19.984 --> 00:06:22.025
And this term right over
here, if we just look at

00:06:22.025 --> 00:06:23.870
it algebraically would
also be, equal to 0,

00:06:23.870 --> 00:06:26.150
so this whole thing
would be equal to 0.

00:06:26.150 --> 00:06:29.160
So someplace in between
x equals 0 and x equals

00:06:29.160 --> 00:06:34.080
10 we should achieve
our maximum volume.

00:06:34.080 --> 00:06:37.120
And before we do it analytically
with a bit of calculus,

00:06:37.120 --> 00:06:38.150
let's do it graphically.

00:06:38.150 --> 00:06:41.100
So I'll get my handy TI-85 out.

00:06:41.100 --> 00:06:43.600
So let me get my TI-85 out.

00:06:43.600 --> 00:06:47.140
And so first let me set
my range appropriately

00:06:47.140 --> 00:06:48.460
before I attempt to graph it.

00:06:48.460 --> 00:06:50.260
So I'll put my graph function.

00:06:50.260 --> 00:06:51.910
Let me set my range.

00:06:51.910 --> 00:06:53.990
So my minimum x-value,
let me make that 0.

00:06:53.990 --> 00:06:55.980
We know that x cannot
be less than 0.

00:06:55.980 --> 00:07:00.190
My maximum x-value, well,
10 seems pretty good.

00:07:00.190 --> 00:07:01.980
My minimum y-value,
this is essentially

00:07:01.980 --> 00:07:03.100
going to be my volume.

00:07:03.100 --> 00:07:04.683
I'm not going to
have negative volume,

00:07:04.683 --> 00:07:06.500
so let me set that equal 0.

00:07:06.500 --> 00:07:08.320
And my maximum
y-value, let's see

00:07:08.320 --> 00:07:10.050
what would be reasonable here.

00:07:10.050 --> 00:07:12.010
I'm just going to
pick some a random x

00:07:12.010 --> 00:07:13.720
and see what type
of a volume I get.

00:07:13.720 --> 00:07:19.540
So if x were to be 5, it
would be 5 times 20 minus 10,

00:07:19.540 --> 00:07:21.650
which is 10.

00:07:21.650 --> 00:07:23.340
So that would be--
did I do that right?

00:07:23.340 --> 00:07:25.320
Yeah, 20 minus 2
times 5, so that

00:07:25.320 --> 00:07:28.620
would be 10, and then times
30 minus 2 times 5, which

00:07:28.620 --> 00:07:29.740
would be 20.

00:07:29.740 --> 00:07:33.560
So it would be 5
times 10 times 20.

00:07:33.560 --> 00:07:36.300
So you'd get a volume
of 1,000 cubic inches.

00:07:36.300 --> 00:07:38.510
And I just randomly
picked the number 5.

00:07:38.510 --> 00:07:41.960
So let me give my maximum
y-value a little higher

00:07:41.960 --> 00:07:44.300
than that just in case to
that isn't the maximum value.

00:07:44.300 --> 00:07:45.466
I just randomly picked that.

00:07:45.466 --> 00:07:48.610
So let's say yMax is 1,500,
and if for whatever reason

00:07:48.610 --> 00:07:50.330
our graph doesn't
fit in there, then

00:07:50.330 --> 00:07:53.290
maybe we can make
our yMax even larger.

00:07:53.290 --> 00:07:55.980
So I think this is going
to be a decent range.

00:07:55.980 --> 00:07:58.170
Now let's actually input
the function itself.

00:07:58.170 --> 00:08:02.250
So our volume is
equal to x times 20

00:08:02.250 --> 00:08:08.930
minus 2x minus 2x
times 30 minus 2x.

00:08:12.870 --> 00:08:14.990
And that looks about right.

00:08:14.990 --> 00:08:16.620
So now I think we can graph it.

00:08:16.620 --> 00:08:19.000
So 2nd, and I want to
select that up there,

00:08:19.000 --> 00:08:21.040
so I'll do Graph.

00:08:21.040 --> 00:08:25.580
And it looks like we
did get the right range.

00:08:25.580 --> 00:08:28.720
So this tells us volume is a
function of x between x is 0

00:08:28.720 --> 00:08:31.730
and x is 10, and it does look
like we hit a maximum point

00:08:31.730 --> 00:08:33.000
right around there.

00:08:33.000 --> 00:08:33.696
So what I'm going
to do is I'm going

00:08:33.696 --> 00:08:35.950
to use the Trace function
to figure out roughly

00:08:35.950 --> 00:08:37.360
what that maximum point is.

00:08:37.360 --> 00:08:39.559
So let me trace this function.

00:08:39.559 --> 00:08:41.070
So I can still go
higher, higher.

00:08:41.070 --> 00:08:41.570
OK.

00:08:41.570 --> 00:08:45.490
So over there my
volume is 1,055.5.

00:08:45.490 --> 00:08:48.990
Then I can get to 1,056.

00:08:48.990 --> 00:08:54.930
So let's see, this was
1,056.20, this is 1,056.24,

00:08:54.930 --> 00:08:57.060
then I go back to 1,055.

00:08:57.060 --> 00:08:59.950
So at least based
on the level of zoom

00:08:59.950 --> 00:09:01.450
that I have my
calculator right now,

00:09:01.450 --> 00:09:03.760
this is a pretty
good approximation

00:09:03.760 --> 00:09:07.650
for the maximum value that my
function actually takes on.

00:09:07.650 --> 00:09:10.530
So it looks like my maximum
value is around 1,056,

00:09:10.530 --> 00:09:15.110
and it happens at
around x equals 3.89.

00:09:15.110 --> 00:09:23.020
So it looks like
my volume at 3.89

00:09:23.020 --> 00:09:28.630
is approximately equal
to 1,056 cubic inches.

00:09:28.630 --> 00:09:33.010
Or you could say that
we hit a maximum when

00:09:33.010 --> 00:09:39.150
x is approximately
equal to 3.89.

00:09:39.150 --> 00:09:41.880
So far, we've just set up
our maximization problem,

00:09:41.880 --> 00:09:44.480
and we've looked
at it graphically.

00:09:44.480 --> 00:09:46.930
In the next video, we'll
try to solve it analytically

00:09:46.930 --> 00:09:49.870
using some of our
calculus tools.


Optimization: box volume (Part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.560
In the last video, we were
able to get a pretty good sense

00:00:04.560 --> 00:00:08.630
about how large of an x we
should cut out of each corner

00:00:08.630 --> 00:00:11.030
in order to maximize our volume.

00:00:11.030 --> 00:00:12.254
And we did this graphically.

00:00:12.254 --> 00:00:13.670
What I want to do
in this video is

00:00:13.670 --> 00:00:16.760
use some of our calculus
tools to see if we can come up

00:00:16.760 --> 00:00:19.054
with the same or maybe
even a better result.

00:00:19.054 --> 00:00:20.970
So to do that, I'm going
to have to figure out

00:00:20.970 --> 00:00:24.000
the critical points of our
volume as a function of x.

00:00:24.000 --> 00:00:26.800
And to do that, I need to take
the derivative of the volume.

00:00:26.800 --> 00:00:27.810
So let me do that.

00:00:27.810 --> 00:00:30.660
And before I even do that,
it'll simplify things

00:00:30.660 --> 00:00:32.950
so I don't have to use the
product rule in some way

00:00:32.950 --> 00:00:34.540
and then have to
simplify that, let

00:00:34.540 --> 00:00:36.830
me just multiply
this expression out.

00:00:36.830 --> 00:00:41.670
So let's rewrite volume as a
function of x is equal to--

00:00:41.670 --> 00:00:44.610
and I'll write it all in yellow.

00:00:44.610 --> 00:00:46.710
So it's going to
be x times-- I'll

00:00:46.710 --> 00:00:48.450
multiply these two
binomials first.

00:00:48.450 --> 00:00:52.310
So 20 times 30 is 600.

00:00:52.310 --> 00:00:58.500
Then I have 20 times negative
2x, which is negative 40x.

00:00:58.500 --> 00:01:03.470
Then I have negative 2x times
30, which is negative 60x.

00:01:03.470 --> 00:01:06.010
And then I have negative
2x times negative 2x,

00:01:06.010 --> 00:01:10.620
which is positive 4x squared.

00:01:10.620 --> 00:01:12.360
So this part over
here simplifies

00:01:12.360 --> 00:01:21.190
and I can change the order to
4x squared minus 100x plus 600.

00:01:21.190 --> 00:01:23.690
I just switched the order
in which I'm writing them.

00:01:23.690 --> 00:01:24.910
So that's that.

00:01:24.910 --> 00:01:27.260
And so I can rewrite
the volume of x

00:01:27.260 --> 00:01:30.635
as being equal to x times all
of this business, which is-- let

00:01:30.635 --> 00:01:32.010
me make sure I
have enough space,

00:01:32.010 --> 00:01:33.509
let me do it a
little higher-- which

00:01:33.509 --> 00:01:39.710
is equal to 4x to the third
power minus 100x squared

00:01:39.710 --> 00:01:41.985
plus 600x.

00:01:41.985 --> 00:01:43.860
And that'll be pretty
straightforward to take

00:01:43.860 --> 00:01:44.980
the derivative.

00:01:44.980 --> 00:01:50.720
So let's say that v prime of
x is going to be equal to-- I

00:01:50.720 --> 00:01:54.280
just have used the power rule
multiple times-- so 4 times 3

00:01:54.280 --> 00:01:55.740
is 12.

00:01:55.740 --> 00:02:00.910
x to the 3 minus 1 power
12x squared minus 200 times

00:02:00.910 --> 00:02:07.770
x to the first power,
which is just x, plus 600.

00:02:07.770 --> 00:02:09.940
And so now we just
have to figure out

00:02:09.940 --> 00:02:11.810
when this is equal to 0.

00:02:11.810 --> 00:02:14.600
So we have to figure
out when 12x squared

00:02:14.600 --> 00:02:19.830
minus 200x plus
600 is equal to 0.

00:02:19.830 --> 00:02:23.090
What x values gets my
derivative to be equal to 0?

00:02:23.090 --> 00:02:26.299
When is my slope equal to 0?

00:02:26.299 --> 00:02:27.840
I could also look
for critical points

00:02:27.840 --> 00:02:29.720
where the derivative
is undefined,

00:02:29.720 --> 00:02:32.640
but this derivative
is defined especially

00:02:32.640 --> 00:02:39.980
throughout my domain of x that
I care about, between 0 and 10.

00:02:39.980 --> 00:02:42.620
So I could try to
factor this or try

00:02:42.620 --> 00:02:43.910
to simplify this a little bit.

00:02:43.910 --> 00:02:45.493
But I'm just going
to cut to the chase

00:02:45.493 --> 00:02:47.810
and try to use the
quadratic formula here.

00:02:47.810 --> 00:02:50.390
So this is, the x's
that satisfy this

00:02:50.390 --> 00:02:54.580
is going to be x is going to
be equal to-- So negative b.

00:02:54.580 --> 00:02:56.580
So it's 200.

00:02:56.580 --> 00:02:59.740
Negative, negative
200 is positive 200.

00:02:59.740 --> 00:03:04.910
Plus or minus the square
root of b squared,

00:03:04.910 --> 00:03:07.340
which is negative 200 squared.

00:03:07.340 --> 00:03:09.812
So I could just write
that as 200 squared.

00:03:09.812 --> 00:03:12.270
Doesn't matter if it's negative
200 squared or 200 squared,

00:03:12.270 --> 00:03:14.080
I'm going to get the same value.

00:03:14.080 --> 00:03:17.510
So let me give myself
some more space.

00:03:17.510 --> 00:03:19.210
So negative 200 squared.

00:03:19.210 --> 00:03:22.320
Well, that's going
to be 4 with 4 0's.

00:03:22.320 --> 00:03:24.620
1, 2, 3, 4.

00:03:24.620 --> 00:03:29.280
So that's going to
be 40,000 minus 4ac.

00:03:29.280 --> 00:03:36.340
So minus 4 times
12, 12 times 600--

00:03:36.340 --> 00:03:40.550
I still didn't give myself
enough space-- times 600.

00:03:40.550 --> 00:03:43.860
All of that over 2 times a.

00:03:43.860 --> 00:03:45.990
So all of that over 24.

00:03:45.990 --> 00:03:49.740
And I'll take out the calculator
again to try to calculate this.

00:03:49.740 --> 00:03:51.970
So let me get out
of graphing mode.

00:03:51.970 --> 00:03:56.970
All right, so first I'll
try when I add the radical.

00:03:56.970 --> 00:04:05.920
So I'm going to get 200 plus
the square root of 40,000.

00:04:05.920 --> 00:04:08.974
I could have just written
that as 200 squared,

00:04:08.974 --> 00:04:09.640
but that's fine.

00:04:09.640 --> 00:04:18.320
40,000 minus 4
times 12 times 600.

00:04:18.320 --> 00:04:24.750
And I get 305, which I
then need to divide by 24.

00:04:24.750 --> 00:04:29.470
Which I'll divide by
24, and I get 12.74.

00:04:29.470 --> 00:04:31.670
So one of my possible x's.

00:04:31.670 --> 00:04:34.959
So it equals 12.74.

00:04:34.959 --> 00:04:36.500
And now let me do
the situation where

00:04:36.500 --> 00:04:39.100
I subtract what I had
in the radical sign.

00:04:39.100 --> 00:04:40.960
So let me get my
calculator back.

00:04:40.960 --> 00:04:43.772
And so now let me
do 200-- I probably

00:04:43.772 --> 00:04:45.730
could have done this
slightly more efficiently,

00:04:45.730 --> 00:04:51.870
but this is fine-- minus the
square root of 40,000-- 1, 2,

00:04:51.870 --> 00:04:59.810
3-- minus 4 times 12 times 600.

00:04:59.810 --> 00:05:01.790
And I get that's
just the numerator.

00:05:01.790 --> 00:05:09.547
And then I'm going to divide
that by 24, and I get 3.92.

00:05:09.547 --> 00:05:10.380
Did I do that right?

00:05:10.380 --> 00:05:15.170
200 minus 40,000 minus 4 times
12 times 600, all of that

00:05:15.170 --> 00:05:16.360
divided by 24.

00:05:16.360 --> 00:05:22.060
My previous answer divided
by 24 gives me 3.92.

00:05:22.060 --> 00:05:27.390
So it's 12.74 or 3.92.

00:05:27.390 --> 00:05:29.710
Now which of these can I use?

00:05:29.710 --> 00:05:36.890
Well x equals 12.74 is outside
of our valid values for x.

00:05:36.890 --> 00:05:42.190
If x was equal to
12.74, the x's would

00:05:42.190 --> 00:05:43.720
start to overlap
with each other.

00:05:43.720 --> 00:05:47.070
So x cannot be 12.74.

00:05:47.070 --> 00:05:51.580
So we get a critical point
at x is equal to 3.92.

00:05:51.580 --> 00:05:53.830
And you could look at the
graph and you could say, oh,

00:05:53.830 --> 00:05:56.550
well look, that looks
like a maximum value.

00:05:56.550 --> 00:05:59.440
But if you didn't have the
graph at your disposal,

00:05:59.440 --> 00:06:01.440
you can then do the
second derivative test

00:06:01.440 --> 00:06:04.370
and say, hey, are we concave
upwards or concave downwards

00:06:04.370 --> 00:06:06.410
when x is equal to 3.92?

00:06:06.410 --> 00:06:08.410
Well, in order to do the
second derivative test,

00:06:08.410 --> 00:06:10.409
you have to figure out
the second derivative is.

00:06:10.409 --> 00:06:12.940
So let's do that.
b prime prime of x

00:06:12.940 --> 00:06:19.960
is going to be equal to 24x,
24 times x to the first,

00:06:19.960 --> 00:06:25.240
minus 200.

00:06:25.240 --> 00:06:26.980
And you can just
look at inspection

00:06:26.980 --> 00:06:30.890
that this number right
over here is less than 4.

00:06:30.890 --> 00:06:33.610
So this thing right over here
is going to be less than 100.

00:06:33.610 --> 00:06:36.170
You subtract 200.

00:06:36.170 --> 00:06:40.140
So we can write the
second derivative at 3.92

00:06:40.140 --> 00:06:41.670
is going to be less than 0.

00:06:41.670 --> 00:06:44.770
You can figure out what the
exact value is if you like.

00:06:44.770 --> 00:06:48.005
So because this is less than
0, we are concave downwards.

00:06:50.900 --> 00:06:52.610
Another way of saying
it is the slope

00:06:52.610 --> 00:06:56.470
is decreasing the entire time.

00:06:56.470 --> 00:06:57.600
Concave downwards.

00:06:57.600 --> 00:06:59.770
When the slope is
decreasing the entire time,

00:06:59.770 --> 00:07:00.980
our shape looks like that.

00:07:00.980 --> 00:07:04.350
The slope could start off
high, lower, lower, gets to 0,

00:07:04.350 --> 00:07:06.070
even lower, lower, lower.

00:07:06.070 --> 00:07:08.410
And we even saw that on
the graph right over here.

00:07:08.410 --> 00:07:10.240
And since it's
concave downwards,

00:07:10.240 --> 00:07:12.860
that implies that our
critical point that

00:07:12.860 --> 00:07:16.820
sits where the interval
is concave downward,

00:07:16.820 --> 00:07:19.435
that critical point
is a local maximum.

00:07:23.920 --> 00:07:27.000
So this is the x value
at which our function

00:07:27.000 --> 00:07:29.520
attains our maximum.

00:07:29.520 --> 00:07:31.820
Now what is that maximum value?

00:07:31.820 --> 00:07:35.060
Well, we could type that back
in into our original expression

00:07:35.060 --> 00:07:37.170
for volume to
figure what that is.

00:07:37.170 --> 00:07:43.350
So let's figure out what the
volume when we get to 3.92

00:07:43.350 --> 00:07:44.090
is equal to.

00:07:44.090 --> 00:07:45.790
What is our maximum volume?

00:07:45.790 --> 00:07:48.900
So get the calculator back out.

00:07:48.900 --> 00:07:50.940
It's obviously roughly 3.92.

00:07:50.940 --> 00:07:52.340
I could use this exact value.

00:07:52.340 --> 00:07:56.010
Actually, I'll just use 3.92
to get a rough sense of what

00:07:56.010 --> 00:07:59.070
our maximum value is,
our maximum volume.

00:07:59.070 --> 00:08:05.640
So it'll be 3.92.

00:08:05.640 --> 00:08:08.420
I'll just use this
expression for the volume

00:08:08.420 --> 00:08:09.810
as a function of x.

00:08:09.810 --> 00:08:29.260
3.92 times 20 minus 2 times
3.92 times 30 minus 2 times 3.92

00:08:29.260 --> 00:08:36.630
gives us-- and we deserve a drum
roll now-- gives us 1,056.3.

00:08:36.630 --> 00:08:42.470
So 1,056.3, which
is a higher volume

00:08:42.470 --> 00:08:44.710
then we got when we just
inspected it graphically.

00:08:44.710 --> 00:08:47.425
We probably could have gotten
a little bit more precise

00:08:47.425 --> 00:08:50.050
if we zoomed in some and then we
would have gotten a little bit

00:08:50.050 --> 00:08:52.080
better of an answer,
but there you have it.

00:08:52.080 --> 00:08:53.830
Analytically, we were
able to actually get

00:08:53.830 --> 00:08:55.270
an even better
answer than we were

00:08:55.270 --> 00:08:58.666
able to do at least on that
first pass graphically.


Inflection points introduction

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:03.150
If you were paying close
attention in the last video,

00:00:03.150 --> 00:00:06.750
an interesting question might
have popped up in your brain.

00:00:06.750 --> 00:00:08.890
We have talked about
the intervals over which

00:00:08.890 --> 00:00:10.609
the function is
concave downwards.

00:00:10.609 --> 00:00:12.150
And then we talked
about the interval

00:00:12.150 --> 00:00:14.820
over which the function
is concave upwards.

00:00:14.820 --> 00:00:17.740
But we see here that there's
a point at which we transition

00:00:17.740 --> 00:00:21.000
from being concave downwards
to concave upwards.

00:00:21.000 --> 00:00:24.290
Before that point, the
slope was decreasing,

00:00:24.290 --> 00:00:26.850
and then the slope
starts increasing.

00:00:26.850 --> 00:00:29.060
The slope was
decreasing and then

00:00:29.060 --> 00:00:31.617
the slope started increasing.

00:00:31.617 --> 00:00:32.950
So that's one way to look at it.

00:00:32.950 --> 00:00:34.741
Right here in our
function we go from being

00:00:34.741 --> 00:00:36.824
concave downwards
to concave upwards.

00:00:36.824 --> 00:00:38.740
When you look at our
derivative at that point,

00:00:38.740 --> 00:00:42.370
our derivative went from
decreasing to increasing.

00:00:42.370 --> 00:00:44.920
And when you look at our second
derivative at that point,

00:00:44.920 --> 00:00:48.160
it went from being
negative to positive.

00:00:48.160 --> 00:00:51.060
So this must have some type of
a special name you're probably

00:00:51.060 --> 00:00:51.560
thinking.

00:00:51.560 --> 00:00:53.230
And you'd be thinking correctly.

00:00:53.230 --> 00:00:57.580
This point at which we
transition from being concave

00:00:57.580 --> 00:01:00.590
downwards to concave upwards,
or the point at which

00:01:00.590 --> 00:01:03.460
our derivative has
a extrema point,

00:01:03.460 --> 00:01:08.090
or the point at which our
second derivative switches signs

00:01:08.090 --> 00:01:10.965
like this, we call it
an inflection point.

00:01:17.450 --> 00:01:19.040
And the most typical
way that people

00:01:19.040 --> 00:01:21.660
think about how could you
test for an inflection point,

00:01:21.660 --> 00:01:23.360
it's a point, well,
conceptually, it's

00:01:23.360 --> 00:01:25.350
where you go from being
a downward opening

00:01:25.350 --> 00:01:26.670
u to an upward opening u.

00:01:26.670 --> 00:01:28.870
Or when you go
from being concave

00:01:28.870 --> 00:01:31.070
downwards to concave upwards.

00:01:31.070 --> 00:01:33.720
But the easiest
test is it's a point

00:01:33.720 --> 00:01:36.700
at which your second
derivative switches signs.

00:01:36.700 --> 00:01:38.930
So in this case, we went
from negative to positive.

00:01:38.930 --> 00:01:40.680
But we could have also
switched from being

00:01:40.680 --> 00:01:41.810
positive to negative.

00:01:41.810 --> 00:01:48.270
So inflection point, your second
derivative f prime prime of x

00:01:48.270 --> 00:01:49.310
switches signs.

00:01:49.310 --> 00:01:51.340
Goes from being
positive to negative

00:01:51.340 --> 00:01:52.720
or negative to positive.

00:01:52.720 --> 00:01:54.260
Switches signs.

00:01:54.260 --> 00:01:56.510
So this is a case where we
went from concave downwards

00:01:56.510 --> 00:01:59.250
to concave upwards.

00:01:59.250 --> 00:02:05.050
If we went from concave
upwards to concave downwards,

00:02:05.050 --> 00:02:11.009
like that, then this inflection
point up until that point,

00:02:11.009 --> 00:02:12.630
the slope was increasing.

00:02:12.630 --> 00:02:15.530
So the second derivative
would to be positive.

00:02:15.530 --> 00:02:18.740
And then the slope
is decreasing,

00:02:18.740 --> 00:02:20.646
so your second derivative
would be negative.

00:02:20.646 --> 00:02:22.020
So here your second
derivative is

00:02:22.020 --> 00:02:24.211
going from positive to negative.

00:02:24.211 --> 00:02:25.710
Here your second
derivative is going

00:02:25.710 --> 00:02:27.020
from negative to positive.

00:02:27.020 --> 00:02:34.230
In either case, you are talking
about an inflection point.


Concavity introduction

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.270
What I have here in yellow is
the graph of y equals f of x.

00:00:04.270 --> 00:00:06.350
Then here in this
mauve color I've

00:00:06.350 --> 00:00:09.310
graphed y is equal to
the derivative of f

00:00:09.310 --> 00:00:10.870
is f prime of x.

00:00:10.870 --> 00:00:12.580
And then here in
blue, I've graphed

00:00:12.580 --> 00:00:15.740
y is equal to the second
derivative of our function.

00:00:15.740 --> 00:00:18.380
So this is the
derivative of this,

00:00:18.380 --> 00:00:21.550
of the first derivative
right over there.

00:00:21.550 --> 00:00:23.830
And we've already seen
examples of how can we

00:00:23.830 --> 00:00:25.345
identify minimum
and maximum points.

00:00:25.345 --> 00:00:27.220
Obviously if we have
the graph in front of us

00:00:27.220 --> 00:00:29.850
it's not hard for a
human brain to identify

00:00:29.850 --> 00:00:31.950
this as a local maximum point.

00:00:31.950 --> 00:00:34.530
The function might take
on higher values later on.

00:00:34.530 --> 00:00:37.540
And to identify this as
a local minimum point.

00:00:37.540 --> 00:00:40.242
The function might take on
the lower values later on.

00:00:40.242 --> 00:00:42.700
But we saw, even if we don't
have the graph in front of us,

00:00:42.700 --> 00:00:45.347
if we were able to take the
derivative of the function

00:00:45.347 --> 00:00:47.180
we might-- or even if
we're not able to take

00:00:47.180 --> 00:00:48.638
the derivative of
the function-- we

00:00:48.638 --> 00:00:51.501
might be able to identify these
points as minimum or maximum.

00:00:51.501 --> 00:00:53.000
The way that we did
it, we said, OK,

00:00:53.000 --> 00:00:55.070
what are the critical
points for this function?

00:00:55.070 --> 00:00:58.360
Well, critical points are
where the function's derivative

00:00:58.360 --> 00:01:00.099
is either undefined or 0.

00:01:00.099 --> 00:01:01.515
This is the
function's derivative.

00:01:01.515 --> 00:01:04.170
It is 0 here and here.

00:01:04.170 --> 00:01:05.810
So we would call
those critical points.

00:01:05.810 --> 00:01:08.530
And I don't see
any points at which

00:01:08.530 --> 00:01:10.590
the derivative is
undefined just yet.

00:01:10.590 --> 00:01:15.590
So we would call here
and here critical points.

00:01:15.590 --> 00:01:19.740
So these are candidate points
at which our function might

00:01:19.740 --> 00:01:21.800
take on a minimum
or a maximum value.

00:01:21.800 --> 00:01:23.550
And the way that we
figured out whether it

00:01:23.550 --> 00:01:25.220
was a minimum or
a maximum value is

00:01:25.220 --> 00:01:29.320
to look at the behavior of the
derivative around that point.

00:01:29.320 --> 00:01:36.320
And over here we saw the
derivative is positive

00:01:36.320 --> 00:01:37.715
as we approach that point.

00:01:41.210 --> 00:01:42.800
And then it becomes negative.

00:01:42.800 --> 00:01:44.800
It goes from being
positive to negative

00:01:44.800 --> 00:01:46.500
as we cross that point.

00:01:46.500 --> 00:01:48.760
Which means that the
function was increasing.

00:01:48.760 --> 00:01:50.679
If the derivative
is positive, that

00:01:50.679 --> 00:01:52.970
means that the function was
increasing as we approached

00:01:52.970 --> 00:01:56.310
that point, and then decreasing
as we leave that point, which

00:01:56.310 --> 00:01:58.544
is a pretty good way
to think about this

00:01:58.544 --> 00:01:59.460
being a maximum point.

00:01:59.460 --> 00:02:00.765
If we're increasing
as we approach it

00:02:00.765 --> 00:02:02.630
and decreasing as we
leave it, then this

00:02:02.630 --> 00:02:06.490
is definitely going
to be a maximum point.

00:02:06.490 --> 00:02:09.330
Similarly, right
over here we see

00:02:09.330 --> 00:02:14.630
that the derivative is negative
as we approach the point, which

00:02:14.630 --> 00:02:17.420
means that the
function is decreasing.

00:02:17.420 --> 00:02:19.780
And we see that the
derivative is positive

00:02:19.780 --> 00:02:20.940
as we exit that point.

00:02:20.940 --> 00:02:22.710
We go from having a
negative derivative

00:02:22.710 --> 00:02:24.530
to a positive
derivative, which means

00:02:24.530 --> 00:02:27.930
the function goes from
decreasing to increasing

00:02:27.930 --> 00:02:30.680
right around that point, which
is a pretty good indication,

00:02:30.680 --> 00:02:33.590
or that is the indication,
that this critical point is

00:02:33.590 --> 00:02:38.710
a point at which the function
takes on a minimum value.

00:02:38.710 --> 00:02:41.150
What I want to do
now is extend things

00:02:41.150 --> 00:02:43.270
by using the idea of concavity.

00:02:46.365 --> 00:02:47.740
And I know I'm
mispronouncing it.

00:02:47.740 --> 00:02:49.740
Maybe it's concavity.

00:02:49.740 --> 00:02:52.630
But thinking about
concavity, we could

00:02:52.630 --> 00:02:55.380
start to look at the second
derivative rather than kind

00:02:55.380 --> 00:02:57.990
of seeing just this transition
to think about whether this

00:02:57.990 --> 00:03:01.280
is a minimum or a maximum point.

00:03:01.280 --> 00:03:03.300
So let's think about
what's happening

00:03:03.300 --> 00:03:06.030
in this first region,
this part of the curve

00:03:06.030 --> 00:03:09.540
up here where it looks like
a arc where it's opening

00:03:09.540 --> 00:03:11.290
downward, where it
looks kind of like an A

00:03:11.290 --> 00:03:13.716
without the cross beam
or an upside down U.

00:03:13.716 --> 00:03:15.090
And then we'll
think about what's

00:03:15.090 --> 00:03:19.950
happening in this kind of upward
opening U part of the curve.

00:03:19.950 --> 00:03:21.790
So over this first
interval, right

00:03:21.790 --> 00:03:23.700
over here, if we start
over here the slope

00:03:23.700 --> 00:03:26.484
is very-- actually let me do
it in a-- actually I'll do it

00:03:26.484 --> 00:03:27.900
in that same color,
because that's

00:03:27.900 --> 00:03:30.210
the same color I used for
the actual derivative.

00:03:30.210 --> 00:03:33.210
The slope is very positive.

00:03:33.210 --> 00:03:36.640
Then it becomes less positive.

00:03:36.640 --> 00:03:39.790
Then it becomes
even less positive.

00:03:39.790 --> 00:03:42.840
It eventually gets to 0.

00:03:42.840 --> 00:03:44.160
Then it keeps decreasing.

00:03:44.160 --> 00:03:47.260
Now it becomes slightly
negative, slightly negative,

00:03:47.260 --> 00:03:49.250
then it becomes
even more negative,

00:03:49.250 --> 00:03:51.200
then it becomes
even more negative.

00:03:51.200 --> 00:03:56.409
And then it looks like it stops
decreasing right around there.

00:03:56.409 --> 00:03:58.450
So the slope stops decreasing
right around there.

00:03:58.450 --> 00:03:59.380
And you see that
in the derivative.

00:03:59.380 --> 00:04:01.380
The slope is decreasing,
decreasing, decreasing,

00:04:01.380 --> 00:04:04.970
decreasing until that point,
and then it starts to increase.

00:04:04.970 --> 00:04:10.920
So this entire section
right over here,

00:04:10.920 --> 00:04:12.575
the slope is decreasing.

00:04:18.550 --> 00:04:21.950
And you see it right over here
when we take the derivative.

00:04:21.950 --> 00:04:26.070
The derivative right over
here, over this entire interval

00:04:26.070 --> 00:04:27.287
is decreasing.

00:04:27.287 --> 00:04:29.620
And we also see that when we
take the second derivative.

00:04:29.620 --> 00:04:31.720
If the derivative
is decreasing, that

00:04:31.720 --> 00:04:33.750
means that the second
derivative, the derivative

00:04:33.750 --> 00:04:35.050
of the derivative, is negative.

00:04:35.050 --> 00:04:38.320
And we see that that
is indeed the case.

00:04:38.320 --> 00:04:43.190
Over this entire interval,
the second derivative

00:04:43.190 --> 00:04:45.821
is indeed negative.

00:04:45.821 --> 00:04:47.570
Now what happens as
we start to transition

00:04:47.570 --> 00:04:51.250
to this upward opening
U part of the curve?

00:04:51.250 --> 00:04:54.150
Well, here the derivative
is reasonably negative.

00:04:54.150 --> 00:04:56.250
It's reasonably
negative right there.

00:04:56.250 --> 00:04:58.570
But then it's still
negative, but it

00:04:58.570 --> 00:05:01.850
becomes less negative and less
negative and less negative,

00:05:01.850 --> 00:05:04.670
less negative and less
negative, and less negative.

00:05:04.670 --> 00:05:05.980
Then it becomes 0.

00:05:05.980 --> 00:05:08.000
It becomes 0 right over here.

00:05:08.000 --> 00:05:11.170
And then it becomes more
and more and more positive.

00:05:11.170 --> 00:05:12.890
And you see that
right over here.

00:05:12.890 --> 00:05:16.720
So over this entire interval,
the slope or the derivative

00:05:16.720 --> 00:05:18.430
is increasing.

00:05:18.430 --> 00:05:24.745
So the slope is increasing.

00:05:24.745 --> 00:05:25.870
And you see this over here.

00:05:25.870 --> 00:05:27.560
Over here the slope is 0.

00:05:27.560 --> 00:05:29.950
The slope of the
derivative is 0.

00:05:29.950 --> 00:05:33.220
The derivative itself isn't
changing right at this moment.

00:05:33.220 --> 00:05:37.089
And then you see that
the slope is increasing.

00:05:37.089 --> 00:05:38.630
And once again, we
can visualize that

00:05:38.630 --> 00:05:40.900
on the second derivative, the
derivative of the derivative.

00:05:40.900 --> 00:05:42.530
If the derivative
is increasing, that

00:05:42.530 --> 00:05:44.680
means the derivative of
that must be positive.

00:05:44.680 --> 00:05:49.240
And it is indeed the case that
the derivative is positive.

00:05:49.240 --> 00:05:53.320
And we have a word for
this downward opening

00:05:53.320 --> 00:05:57.640
U and this upward opening U.
We call this concave downwards.

00:06:02.120 --> 00:06:03.560
Let me make this clear.

00:06:03.560 --> 00:06:07.920
Concave downwards.

00:06:07.920 --> 00:06:10.030
And we call this
concave upwards.

00:06:13.400 --> 00:06:15.330
So let's review
how we can identify

00:06:15.330 --> 00:06:18.670
concave downward intervals
and concave upwards intervals.

00:06:18.670 --> 00:06:25.840
So if we're talking
about concave downwards,

00:06:25.840 --> 00:06:27.780
we see several things.

00:06:27.780 --> 00:06:29.340
We see that the
slope is decreasing.

00:06:37.590 --> 00:06:41.430
Which is another
way of saying that f

00:06:41.430 --> 00:06:47.290
prime of x is decreasing.

00:06:51.720 --> 00:06:54.840
Which is another way of saying
that the second derivative must

00:06:54.840 --> 00:06:55.590
be negative.

00:06:55.590 --> 00:06:58.090
If the first derivative
is decreasing,

00:06:58.090 --> 00:07:00.460
the second derivative
must be negative,

00:07:00.460 --> 00:07:03.350
which is another way of saying
that the second derivative

00:07:03.350 --> 00:07:08.160
over that interval
must be negative.

00:07:08.160 --> 00:07:11.010
So if you have a negative
second derivative,

00:07:11.010 --> 00:07:14.220
then you are in a concave
downward interval.

00:07:14.220 --> 00:07:17.480
Similarly-- I have
trouble saying

00:07:17.480 --> 00:07:21.640
that word-- let's think
about concave upwards,

00:07:21.640 --> 00:07:25.630
where you have an upward
opening U. Concave upwards.

00:07:25.630 --> 00:07:28.510
In these intervals, the
slope is increasing.

00:07:28.510 --> 00:07:31.030
We have a negative slope, less
negative, less negative, 0,

00:07:31.030 --> 00:07:34.040
positive, more positive, more
positive, even more positive.

00:07:34.040 --> 00:07:37.780
So slope is increasing.

00:07:43.240 --> 00:07:50.690
Which means that the derivative
of the function is increasing.

00:07:50.690 --> 00:07:52.680
And you see that
right over here.

00:07:52.680 --> 00:07:56.240
This derivative is
increasing in value,

00:07:56.240 --> 00:08:00.150
which means that the second
derivative over an interval

00:08:00.150 --> 00:08:03.490
where we are concave upwards
must be greater than 0.

00:08:03.490 --> 00:08:05.450
If the second derivative
is greater than 0,

00:08:05.450 --> 00:08:06.950
that means that the
first derivative

00:08:06.950 --> 00:08:09.310
is increasing, which means
that the slope is increasing.

00:08:09.310 --> 00:08:14.630
We are in a concave
upward interval.

00:08:14.630 --> 00:08:18.000
Now given all of these
definitions that we've just

00:08:18.000 --> 00:08:20.150
given for concave downwards
and concave upwards,

00:08:20.150 --> 00:08:22.210
can we come up with
another way of identifying

00:08:22.210 --> 00:08:24.820
whether a critical
point is a minimum point

00:08:24.820 --> 00:08:26.490
or a maximum point?

00:08:26.490 --> 00:08:28.420
Well, if you have
a maximum point,

00:08:28.420 --> 00:08:32.500
if you have a critical
point where the function is

00:08:32.500 --> 00:08:36.130
concave downwards, then you're
going to be at a maximum point.

00:08:36.130 --> 00:08:38.210
Concave downwards, let's
just be clear here,

00:08:38.210 --> 00:08:42.263
means that it's
opening down like this.

00:08:42.263 --> 00:08:44.179
And when we're talking
about a critical point,

00:08:44.179 --> 00:08:46.304
if we're assuming it's
concave downwards over here,

00:08:46.304 --> 00:08:48.784
we're assuming differentiability
over this interval.

00:08:48.784 --> 00:08:50.200
And so the critical
point is going

00:08:50.200 --> 00:08:52.350
to be one where the slope is 0.

00:08:52.350 --> 00:08:54.810
So it's going to be that
point right over there.

00:08:54.810 --> 00:08:56.640
So if you're concave
downwards and you

00:08:56.640 --> 00:09:02.250
have a point where f prime of,
let's say, a is equal to 0,

00:09:02.250 --> 00:09:04.730
then we have a
maximum point at a.

00:09:11.500 --> 00:09:14.210
And similarly, if
we're concave upwards,

00:09:14.210 --> 00:09:17.340
that means that our function
looks something like this.

00:09:17.340 --> 00:09:20.210
And if we found a point,
obviously a critical point

00:09:20.210 --> 00:09:22.600
could also be where the
function is not defined,

00:09:22.600 --> 00:09:24.800
but if we're assuming
that our first derivative

00:09:24.800 --> 00:09:26.510
and second derivative
is defined here,

00:09:26.510 --> 00:09:28.260
then the critical point
is going to be one

00:09:28.260 --> 00:09:31.380
where the first derivative
is going to be 0.

00:09:31.380 --> 00:09:35.090
So f prime of a is equal to 0.

00:09:35.090 --> 00:09:38.130
And if f prime of
a is equal to 0

00:09:38.130 --> 00:09:40.900
and if we're concave
upwards in the interval

00:09:40.900 --> 00:09:44.250
around a, so if the second
derivative is greater than 0,

00:09:44.250 --> 00:09:46.010
then it's pretty
clear, you see here,

00:09:46.010 --> 00:09:53.610
that we are dealing with
a minimum point at a.


Analyzing a function with its derivative

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.010
We've got the function f of x
is equal to x to the third power

00:00:04.010 --> 00:00:06.270
minus 12x plus 2.

00:00:06.270 --> 00:00:08.140
And what I want to
do in this video

00:00:08.140 --> 00:00:09.770
is think about at
what points does

00:00:09.770 --> 00:00:13.559
my function f take on
minimum or maximum values?

00:00:13.559 --> 00:00:15.600
And to figure that out I
have to first figure out

00:00:15.600 --> 00:00:17.730
what are the critical
points for my function f.

00:00:17.730 --> 00:00:19.470
And then which of
those critical points

00:00:19.470 --> 00:00:22.250
do we achieve a minimum
or maximum value?

00:00:22.250 --> 00:00:23.790
And to determine
the critical points

00:00:23.790 --> 00:00:25.706
we have to find the
derivative of our function

00:00:25.706 --> 00:00:27.710
because our critical
points are just

00:00:27.710 --> 00:00:31.230
the point at which our
derivative is either equal to 0

00:00:31.230 --> 00:00:32.860
or undefined.

00:00:32.860 --> 00:00:34.882
So the derivative of this
thing right over here,

00:00:34.882 --> 00:00:37.090
we're just going to use the
power rule several times,

00:00:37.090 --> 00:00:39.215
and then I guess you can
call it the constant rule.

00:00:39.215 --> 00:00:42.140
But the derivative of x to
the third is 3x squared.

00:00:42.140 --> 00:00:44.590
Derivative of negative
12x is negative 12.

00:00:44.590 --> 00:00:46.090
And the derivative
of a constant, it

00:00:46.090 --> 00:00:47.464
doesn't change
with respect to x,

00:00:47.464 --> 00:00:49.649
so it's just going
to be equal to 0.

00:00:49.649 --> 00:00:51.440
So we're going to get
a critical point when

00:00:51.440 --> 00:00:54.430
this thing right over
here, for some value of x

00:00:54.430 --> 00:00:56.540
is either undefined or 0.

00:00:56.540 --> 00:00:59.121
Well this thing is defined
for all values of x.

00:00:59.121 --> 00:01:01.370
So the only places we're
going to find critical points

00:01:01.370 --> 00:01:02.930
is when this thing
is equal to 0.

00:01:02.930 --> 00:01:04.800
So let's set it equal to 0.

00:01:04.800 --> 00:01:09.090
When does 3x squared
minus 12 equal 0?

00:01:09.090 --> 00:01:11.340
So let's add 12 to both sides.

00:01:11.340 --> 00:01:14.460
You get 3x squared
is equal to 12.

00:01:14.460 --> 00:01:16.930
Divide both sides by 3.

00:01:16.930 --> 00:01:19.680
You get x squared is equal to 4.

00:01:19.680 --> 00:01:23.790
Well this is going to
happen when x is equal to 2

00:01:23.790 --> 00:01:28.980
and x is equal to negative 2.

00:01:28.980 --> 00:01:33.190
Just to be clear, f of
2, or let me be clear,

00:01:33.190 --> 00:01:38.160
f prime of 2, you get 3 times 4
minus 12, which is equal to 0.

00:01:38.160 --> 00:01:43.640
And f prime negative 2 is
also, same exact reason,

00:01:43.640 --> 00:01:45.370
is also equal to 0.

00:01:45.370 --> 00:01:48.130
So we can say-- and I'll
switch colors here--

00:01:48.130 --> 00:01:58.370
that f has critical
points at x equals 2

00:01:58.370 --> 00:02:01.450
and x equals negative 2.

00:02:01.450 --> 00:02:02.450
Well that's fair enough.

00:02:02.450 --> 00:02:05.590
But we still don't know
whether the function takes

00:02:05.590 --> 00:02:08.720
on a minimum values at
those points, maximum values

00:02:08.720 --> 00:02:10.864
of those points, or neither.

00:02:10.864 --> 00:02:12.530
To figure that out
we have to figure out

00:02:12.530 --> 00:02:16.305
whether the derivative changes
signs around these points.

00:02:16.305 --> 00:02:18.180
So let's actually try
to graph the derivative

00:02:18.180 --> 00:02:19.480
to think about this.

00:02:19.480 --> 00:02:20.560
So let's graph.

00:02:20.560 --> 00:02:22.989
I'll draw an axis
right over here.

00:02:22.989 --> 00:02:24.530
I'll do it down here
because maybe we

00:02:24.530 --> 00:02:27.860
can use that information
later on to graph f of x.

00:02:27.860 --> 00:02:33.710
So let's say this is my x-axis.

00:02:33.710 --> 00:02:38.960
This is my y-axis.

00:02:38.960 --> 00:02:41.940
And so we have critical points
at x is equal to positive 2.

00:02:41.940 --> 00:02:44.190
So it's 1, 2.

00:02:44.190 --> 00:02:50.620
And x is equal to negative 2,
1, 2. x is equal to negative 2.

00:02:50.620 --> 00:02:53.250
So what does this
derivative look

00:02:53.250 --> 00:02:55.280
like if we wanted to graph it?

00:02:55.280 --> 00:02:59.990
Well we have when x is equal
to 0 for the derivative

00:02:59.990 --> 00:03:01.540
we're at negative 12.

00:03:01.540 --> 00:03:04.780
So this is the point y
is equal to negative 12.

00:03:04.780 --> 00:03:08.650
So this is, we're graphing
y is equal to f prime of x.

00:03:08.650 --> 00:03:10.635
So it looks something like this.

00:03:15.040 --> 00:03:17.900
These are obviously the
0's of our derivative.

00:03:17.900 --> 00:03:22.295
So it has to move up to cross
the x-axis there and over here.

00:03:25.980 --> 00:03:28.660
So what is the
derivative doing at each

00:03:28.660 --> 00:03:30.480
of these critical points?

00:03:30.480 --> 00:03:35.170
Well over here our derivative
is crossing from being positive,

00:03:35.170 --> 00:03:40.150
we have a positive derivative,
to being a negative derivative.

00:03:40.150 --> 00:03:42.980
So we're crossing from being
a positive derivative to being

00:03:42.980 --> 00:03:45.620
a negative derivative,
that was our criteria

00:03:45.620 --> 00:03:47.750
for a critical point
to be a maximum point.

00:03:51.580 --> 00:03:57.590
Over here we're crossing
from a negative derivative

00:03:57.590 --> 00:04:00.170
to a positive
derivative, which is

00:04:00.170 --> 00:04:04.470
our criteria for a critical
point for the function

00:04:04.470 --> 00:04:07.210
to have a minimum value
at a critical point.

00:04:07.210 --> 00:04:08.600
So a minimum.

00:04:08.600 --> 00:04:12.710
And I just want to make sure
we have the correct intuition.

00:04:12.710 --> 00:04:15.780
If our function, if some
function is increasing

00:04:15.780 --> 00:04:20.980
going into some point,
and at that point

00:04:20.980 --> 00:04:22.770
we actually have
a derivative 0--

00:04:22.770 --> 00:04:24.720
the derivative could
also be undefined--

00:04:24.720 --> 00:04:27.130
but we have a derivative
of 0 and then the function

00:04:27.130 --> 00:04:32.040
begins decreasing, that's why
this would be a maximum point.

00:04:32.040 --> 00:04:35.200
Similarly, if we have a
situation where the function is

00:04:35.200 --> 00:04:38.086
decreasing going into a point,
the derivative is negative.

00:04:38.086 --> 00:04:39.960
Remember this is the
graph of the derivative.

00:04:39.960 --> 00:04:41.410
Let me make this clear.

00:04:41.410 --> 00:04:44.930
This is the graph of y is equal
to not f of x, but f prime

00:04:44.930 --> 00:04:45.980
of x.

00:04:45.980 --> 00:04:48.580
So if we have a situation
we're going into the point,

00:04:48.580 --> 00:04:52.521
the function has
a negative slope,

00:04:52.521 --> 00:04:55.020
we see we have a negative slope
here-- so the function might

00:04:55.020 --> 00:04:56.552
look something like this.

00:04:56.552 --> 00:04:58.260
And then right at this
point the function

00:04:58.260 --> 00:05:00.350
is either undefined
or has 0 slope.

00:05:00.350 --> 00:05:02.770
So in this case it has 0 slope.

00:05:02.770 --> 00:05:07.070
And then after that point,
let me do it right under it.

00:05:07.070 --> 00:05:10.730
So going into it we
have a negative slope.

00:05:10.730 --> 00:05:12.650
And then right over
here we have a 0 slope.

00:05:12.650 --> 00:05:15.080
Which I could draw it
even better than that.

00:05:15.080 --> 00:05:17.810
So if we were to imagine
going into it we have

00:05:17.810 --> 00:05:21.080
a negative slope, right at
that point we have a 0 slope,

00:05:21.080 --> 00:05:23.510
and then we have
a positive slope.

00:05:23.510 --> 00:05:25.070
So the function
begins increasing.

00:05:25.070 --> 00:05:28.952
That's why we say we have a
minimum point right over there.

00:05:28.952 --> 00:05:30.410
So what I did it
right over here is

00:05:30.410 --> 00:05:33.160
to try to conceptualize what
the function itself could look

00:05:33.160 --> 00:05:35.700
like given the derivative,
in this case switching

00:05:35.700 --> 00:05:38.240
from a positive derivative
to a negative derivative,

00:05:38.240 --> 00:05:40.460
across that critical
point, or going

00:05:40.460 --> 00:05:42.700
from a negative derivative
to a positive derivative.

00:05:42.700 --> 00:05:45.120
That's why this is the
criteria for a maximum point,

00:05:45.120 --> 00:05:47.350
this is the criteria
for a minimum point.

00:05:47.350 --> 00:05:49.830
Well with that out of the
way, can we use this intuition

00:05:49.830 --> 00:05:51.920
that we just talked
about to at least try

00:05:51.920 --> 00:05:54.830
to sketch the graph of f of x?

00:05:54.830 --> 00:05:56.000
So let's try to do it.

00:05:59.692 --> 00:06:01.150
And it's just going
to be a sketch,

00:06:01.150 --> 00:06:02.584
it's not going to be very exact.

00:06:02.584 --> 00:06:04.000
But at least it'll
give us a sense

00:06:04.000 --> 00:06:07.010
of the shape of what
f of x looks like.

00:06:07.010 --> 00:06:09.140
So my best attempt.

00:06:09.140 --> 00:06:13.020
So it might not be drawn
completely to scale.

00:06:13.020 --> 00:06:16.520
So it's my x-axis,
this is my y-axis.

00:06:16.520 --> 00:06:21.170
We know we have a critical point
at x is equal to positive 2.

00:06:21.170 --> 00:06:28.340
And we have a critical point
at x is equal to negative 2.

00:06:28.340 --> 00:06:30.900
We know just from inspection
that the y-intercept right

00:06:30.900 --> 00:06:32.550
here, if the graph
of y is equal to f

00:06:32.550 --> 00:06:35.580
of x, when x is 0 f of x is 2.

00:06:35.580 --> 00:06:40.371
So we're going to
hit right over--

00:06:40.371 --> 00:06:41.870
I don't want to
draw this completely

00:06:41.870 --> 00:06:44.860
to the same scale as the x-axis.

00:06:44.860 --> 00:06:48.510
So let's say that this
is 2 right over here.

00:06:48.510 --> 00:06:50.570
So this is where
we're going to cross.

00:06:50.570 --> 00:06:53.130
This is going to
be our y-intercept.

00:06:53.130 --> 00:06:55.760
And so we said
already that we have

00:06:55.760 --> 00:06:58.960
a maximum point at x
is equal to negative 2.

00:06:58.960 --> 00:07:06.200
So what is f of negative 2?
f of negative 2 is equal to 8

00:07:06.200 --> 00:07:07.640
or negative 8,
let me be careful.

00:07:07.640 --> 00:07:09.080
It's negative 8.

00:07:09.080 --> 00:07:11.640
And then we're going to have
12 times negative 2, which

00:07:11.640 --> 00:07:12.810
is negative 24.

00:07:12.810 --> 00:07:14.450
But then we're going to add it.

00:07:14.450 --> 00:07:16.620
So we're subtracting
negative 24.

00:07:16.620 --> 00:07:20.080
So this is plus 24.

00:07:20.080 --> 00:07:22.930
And then we finally have plus 2.

00:07:22.930 --> 00:07:28.450
So negative 8 plus 24 plus 2,
so that's going to be negative,

00:07:28.450 --> 00:07:32.340
let's see, negative 8 plus
24 is 16, plus 2 is 18.

00:07:32.340 --> 00:07:34.770
So f of negative
2 is equal to 18.

00:07:34.770 --> 00:07:36.630
And I'm not drawing it
completely to scale.

00:07:36.630 --> 00:07:39.830
But let's say that this
is 18 right over here.

00:07:39.830 --> 00:07:41.730
So this is the function.

00:07:41.730 --> 00:07:44.920
This is the point
negative 2 comma 18.

00:07:44.920 --> 00:07:47.060
And we know that
it's a maximum point.

00:07:47.060 --> 00:07:50.470
The derivative going into
that point is negative.

00:07:50.470 --> 00:07:54.740
The derivative going into that
point is negative-- sorry,

00:07:54.740 --> 00:07:57.070
the derivative going into
that point is positive.

00:07:57.070 --> 00:08:00.110
So we are increasing.

00:08:00.110 --> 00:08:01.620
The slope is positive.

00:08:01.620 --> 00:08:03.450
And then after we
cross that point,

00:08:03.450 --> 00:08:06.900
the slope becomes negative.

00:08:06.900 --> 00:08:13.000
The derivative cross the x-axis,
the slope becomes negative.

00:08:13.000 --> 00:08:15.070
I actually want to
use that same color.

00:08:15.070 --> 00:08:20.140
It looks like this.

00:08:20.140 --> 00:08:22.500
And then, of
course, the graph is

00:08:22.500 --> 00:08:24.510
going to cross the--
it's going to have

00:08:24.510 --> 00:08:27.140
a y-intercept,
something like that.

00:08:27.140 --> 00:08:31.310
And then, as we
approach 2, we are

00:08:31.310 --> 00:08:33.690
approaching another
critical point.

00:08:33.690 --> 00:08:35.830
Now what is f of 2?

00:08:35.830 --> 00:08:44.550
f of 2 is going to be equal
to positive 8 minus 24 plus 2.

00:08:44.550 --> 00:08:47.740
So this is 10 minus 24, which
is equal to negative 14.

00:08:47.740 --> 00:08:50.540
So let's say this is the point
negative 14 right over here.

00:08:50.540 --> 00:08:52.040
Actually I can draw
it a little bit.

00:08:52.040 --> 00:08:54.160
Let's say this is negative 14.

00:08:54.160 --> 00:08:56.870
So this is f of 2
right over there.

00:08:56.870 --> 00:08:59.810
And we saw already that
the slope is negative

00:08:59.810 --> 00:09:00.800
as we approach it.

00:09:00.800 --> 00:09:05.420
So our function is
decreasing as we approach it.

00:09:05.420 --> 00:09:07.160
And then right there
the slope is 0.

00:09:07.160 --> 00:09:08.700
We figured that
out earlier, that's

00:09:08.700 --> 00:09:10.890
how we identified it
being a critical point.

00:09:10.890 --> 00:09:13.700
And then the slope is
increasing after that,

00:09:13.700 --> 00:09:15.240
the derivative is positive.

00:09:15.240 --> 00:09:17.850
The slope is increasing.

00:09:17.850 --> 00:09:21.700
So this is our
sketch of what f of x

00:09:21.700 --> 00:09:24.770
could look like given that
these are the critical points.

00:09:24.770 --> 00:09:30.320
And we were able to identify
2 as a minimum point.

00:09:30.320 --> 00:09:31.870
So this was a minimum value.

00:09:31.870 --> 00:09:35.430
The function takes on a minimum
value when x is equal to 2.

00:09:35.430 --> 00:09:37.310
And the function took
on a maximum value

00:09:37.310 --> 00:09:41.510
when x was equal to negative 2.