## anonymous 4 years ago Let $$f$$ be a function satisfying $$f(x+y) = f(x) + f(y) \forall x,y$$ and if $$f(x) = x^2 g(x)$$ where $$g(x)$$ is a continuous function, then find $$f'(x)$$.

1. anonymous

i am going to make a guess, that $f'(x)=c$ some constant. not sure what that has to do with $f(x)=x^2g(x)$ but if i recall correctly the only function satisfying the first condition is $f(x)=cx$

2. anonymous

$f'(x)=2xg(x)+x^2g'(x)$

3. anonymous

@zed that is true for any f, right?

4. anonymous

i don't think there are any functions that satisfy $f(x+y)=f(x)+f(y)$ other than constant multiples

5. anonymous

Here, these are the options 1. g'(x) 2. g(0) 3. g(0) + g'(x) 3. 0

6. anonymous

multiple choice?

7. Zarkon

looks like zero to me

8. anonymous

yeah, but i am clueless

9. anonymous

really? why. i am fairly certain $f'(x)=c$ a constant

10. Zarkon

us the definition of the derivative

11. Zarkon

*use

12. Zarkon

(f(x+h)-f(x))/h (f(x)+f(h)-f(x))/h =f(h)/h

13. Zarkon

$f(h)=h^2g(h)$

14. Zarkon

$\frac{h^2g(h)}{h}=hg(h)$ take limit as h goes to zero...use squeeze theorem