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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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A hint: For a given $c\in[0,1]$ compute $f(0)$ and $f(1)$ by means of a Taylor expansion at $c$ (with Lagrange remainder term) and draw conclusions.

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    How did you use the assumption $f(0)=f(1)=0\thinspace$?2011-01-09