Problem: Given that $f$ is differentiable at $[0,1]$ and $f(0)=f(1)=0$. If $ \forall x\in (0,1)$ |f''(x)|\leq A show that $\forall x \in [0,1]$ |f'(x)| \leq (A/2) .
My attempt was to to develop a Taylor series for $f(x)$ and f'(x) around point $c \in (0,1)$ where f'(c)=0.