Mathematics
OpenStudy (anonymous):

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 $$f'(x)$$.

OpenStudy (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 $f(x)=cx$

OpenStudy (anonymous):

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

OpenStudy (anonymous):

@zed that is true for any f, right?

OpenStudy (anonymous):

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

OpenStudy (anonymous):

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

OpenStudy (anonymous):

multiple choice?

OpenStudy (zarkon):

looks like zero to me

OpenStudy (anonymous):

yeah, but i am clueless

OpenStudy (anonymous):

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

OpenStudy (zarkon):

us the definition of the derivative

OpenStudy (zarkon):

*use

OpenStudy (zarkon):

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

OpenStudy (zarkon):

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

OpenStudy (zarkon):

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

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