Maximum of a Function Bounded by Average of Integral and Integral of Derivative [duplicate]
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Observe that $$\int_{a}^{b} |f'| \geq \max_{s,t \in [a,b]} \left|\int_{t}^{s} f'\right| =\max_{s,t \in [a,b]} |f(s)-f(t)| \geq \max_{s,t \in [a,b]} |f(s)|-|f(t)| = \max_{s \in [a,b]} |f(s)| - \min_{s \in [a,b]} |f(s)|$$
and $$\frac{1}{b-a} \left|\int_{a}^{b} f\right| = |f(c)|$$ for some $c \in [a,b]$ by the mean value theorem.
The result follows from the fact that $|f(c)| \geq \min_{s \in [a,b]} |f(s)|$.