Continuous and differentiability

[Joe20]

Hello,

I have attached the question and the steps worked out. I am not sure if my steps are correctly. Need advise on that.
Next, I am not sure how to show f''(0) exist or not. Thanks in advance!

 

[Opalg]

You have shown that $f$ is differentiable at $0$ and that $f'(0) = 0$. For $x\ne0$ you can use the usual rules for differentiation, to see that $f'(x) = 2x$ if $x>0$ and $f'(x) = -2x$ if $x<0$. Therefore \[f'(x) = \begin{cases}2x&\text{when }x\geqslant0,\\-2x&\text{when }x<0.\end{cases}\] Now you have to decide whether that function is differentiable at $x=0$.