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 $f'(x)$.

Question

anonymous · Answer

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 
$$f(x)=cx$$

anonymous · Answer

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

anonymous · Answer

@zed that is true for any f, right?

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

multiple choice?

zarkon · Answer

looks like zero to me

anonymous · Answer

yeah, but i am clueless

anonymous · Answer

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

zarkon · Answer

us the definition of the derivative

zarkon · Answer

*use

zarkon · Answer

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

zarkon · Answer

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

zarkon · Answer

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