TITLE: How do I find $f(0)$, $f'(0)$, and $f'(x)$ given $f(x+y)=f(x)+f(y)+x^2y+xy^2$ and $\lim_{x\to0}\frac{f(x)}{x}=1$?
QUESTION [7 upvotes]: How can I find $f(0)$, $f'(0)$, and $f'(x)$ given that
$f(x+y)=f(x)+f(y)+x^2y+xy^2$
and
$\lim_{x\to0}\frac{f(x)}{x}=1$.

REPLY [8 votes]: Let $x=y=0$ we get $f(0)=0$
We have

$$f'(x)=\lim_{y\to0}\frac{f(x+y)-f(x)}{y}=\lim_{y\to0}\frac{f(y)}{y}+x^2+xy=1+x^2$$