Let $f \in C^2[0,2]$ and $|f(x)| \le 1, |f''(x)|\le1$ for all $x \in[0,1]$
Prove that $|f'(x)|\le2 $ for all $x \in [0,2]$
I tried Taylor expansion to evaluate $|f'(x)|$, but it seems to work only when $x\in[0,1]$. I'm confused when it comes to $x \in [1,2]$
Any help will be appreciated!
