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Putnam 2007 B2

Source:

December 3, 2007
Putnamcalculusderivativeintegrationsymmetryfunctionalgebra

Problem Statement

Suppose that f:[0,1]→R f: [0,1]\to\mathbb{R} has a continuous derivative and that \int_0^1f(x)\,dx\equal{}0. Prove that for every α∈(0,1), \alpha\in(0,1), ∣∫0αf(x) dx∣≤18max⁡0≤x≤1∣f′(x)∣ \left|\int_0^{\alpha}f(x)\,dx\right|\le\frac18\max_{0\le x\le 1}|f'(x)|
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