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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 $f'(c)=0$.

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