1. ## Challenge problem.

Hello.
I have this problem, I stucked.
I did not post it in the challenge sub-forum; because I do not know the solution.

Problem: Suppose $f$ is a function satisfies the equation $f(x+y)=f(x)+f(y)+x^2y+xy^2$ for all number $x$ and $y$. Suppose also that: $\lim_{x\to 0} \frac{f(x)}{x}=1$.
• (a) Find $f(0)$.
• (b) Find $f'(0)$.
• (c) Find $f'(x)$.

2. Originally Posted by Ted
Hello.
I have this problem, I stucked.
I did not post it in the challenge sub-forum; because I do not know the solution.

Problem: Suppose $f$ is a function satisfies the equation $f(x+y)=f(x)+f(y)+x^2y+xy^2$ for all number $x$ and $y$. Suppose also that: $\lim_{x\to 0} \frac{f(x)}{x}=1$.
• (a) Find $f(0)$.
• (b) Find $f'(0)$.
• (c) Find $f'(x)$.
(a) $f(0+0) = f(0) + f(0) + 0^2 \cdot 0 + 0 \cdot 0^2
$

$f(0) = 2f(0)$

$f(0) = 0$

(b) $f'(0) = \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0}
$

$f'(0) = \lim_{x \to 0} \frac{f(x)}{x} = 1$

(c) $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

$f'(x) = \lim_{h \to 0} \frac{f(x) + f(h) + x^2h + xh^2 - f(x)}{h}$

$f'(x) = \lim_{h \to 0} \frac{f(h) + x^2h + xh^2}{h}
$

$f'(x) = \lim_{h \to 0} \left[\frac{f(h)}{h} + \frac{h(x^2 + xh)}{h}\right]
$

$f'(x) = \lim_{h \to 0} \left[\frac{f(h)}{h} + (x^2 + xh) \right]$

$f'(x) = 1 + x^2$