How do I find $f(0)$, $f'(0)$, and $f'(x)$ given $f(x+y)=f(x)+f(y)+x^2y+xy^2$ and $\lim_{x\to0}\frac{f(x)}{x}=1$?
- Let $x=y=0$ we get $f(0)=0$
- We have
$$f'(x)=\lim_{y\to0}\frac{f(x+y)-f(x)}{y}=\lim_{y\to0}\frac{f(y)}{y}+x^2+xy=1+x^2$$
How do I find $f(0)$, $f'(0)$, and $f'(x)$ given $f(x+y)=f(x)+f(y)+x^2y+xy^2$ and $\lim_{x\to0}\frac{f(x)}{x}=1$?
- Let $x=y=0$ we get $f(0)=0$
- We have
$$f'(x)=\lim_{y\to0}\frac{f(x+y)-f(x)}{y}=\lim_{y\to0}\frac{f(y)}{y}+x^2+xy=1+x^2$$