Answer
$f(0) = 0$
$f'(0) = 0$
Work Step by Step
$\vert f(x) \vert \leq x^2~~$ for all $x$
We can find $f(0)$:
$\vert f(0) \vert \leq 0^2 = 0$
Therefore, $f(0) = 0$
We can find $f'(0)$:
$\vert f'(0) \vert = \vert \lim\limits_{x \to 0} \frac{f(x)-f(0)}{x-0}\vert$
$\vert f'(0) \vert = \vert \lim\limits_{x \to 0} \frac{f(x)}{x}\vert \leq \vert \lim\limits_{x \to 0} \frac{x^2}{x}\vert = \vert \lim\limits_{x \to 0} ~x \vert = 0$
Therefore, $f'(0) = 0$
