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Putnam 1973 B4

Source: Putnam 1973

May 25, 2022
PutnaminequalitiesintegrationIntegral

Problem Statement

(a) On [0,1][0, 1], let f(x)f(x) have a continuous derivative satisfying 0<f′(x)≤10 <f'(x) \leq1. Also suppose that f(0)=0.f(0) = 0. Prove that (∫01f(x)  dx)2≥∫01f(x)3  dx. \left( \int_{0}^{1} f(x)\; dx \right)^{2} \geq \int_{0}^{1} f(x)^{3}\; dx. (b) Show an example in which equality occurs.
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