Q:

Find a function s.t. $f(0), f'(0)$ exists but $f''(0)$ doesn't

Find a function $f(x)$ such that $f(0)$ exists and $f'(0)$ exists, but $f''(0)$ does not exist. 

I have no idea how to solve this problem, I've tried many times. 

A:

Take
$f(x)=x^2$ if $x \ge 0$ and $f(x)=-x^2$ if $x <0$.
Show that $f'(x)=2|x|$  for all $x$. Hence $f'$ is not differentiable at $x=0$.

