What is the way to show the following derivative problem? If $f$ is function twice differentiable with $|f''(x)|<1, x\in [0,1]$ and $f(0)=f(1)$, then $|f'(x)|<1$ for all $x\in [0,1]$
I have tried with Rolle's theorem, but fail

Hint: For some $c\in[0,1]$, $f'(c)=0$ (can you see why?).  Now apply the mean value theorem to $f'$ to bound $f'(x)$ for any other $x\in[0,1]$.
