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Inequality of function with integral and derivative

Source: VJIMC 2024, Category I, Problem 1

April 14, 2024
inequalitiesfunctioncalculusintegrationderivative

Problem Statement

Let f:R→Rf:\mathbb{R} \to \mathbb{R} be a continuously differentiable function. Prove that ∣f(1)−∫01f(x)dx∣≤12max⁡x∈[0,1]∣f′(x)∣.\left\vert f(1)-\int_0^1 f(x) dx\right\vert \le \frac{1}{2} \max_{x \in [0,1]} \vert f'(x)\vert.
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