Suppose a function $f$ satisfies the equation $f(x+y)=f(x)+f(y)+xy^2+yx^2$ for all real numbers $x$ and $y$, and $\lim \limits _{x\rightarrow 0}\dfrac{f(x)}{x}=1$. Find 1.$f(0)$, 2.$f^\prime (0)$, 3.$f^\prime (x)$
In my solution I started with number 3, I have $f^\prime (x)=1+x^2$. To answer number 2, I just substituted and got $f^\prime (0)=1$. Now coming to number 1, I expected $f(0)$ to be $x$, but using the given equation, I get $f(0)=0$ which contradicts the solution to number 2. Where did I go wrong?
There is no contradiction; $f(0)$ cannot be a variable $x$. $f(0)$ is indeed $0$.