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Cauchy like integral inequality

Source: 2016 South Korea USCM P7

August 16, 2020
inequalitiesholder inequalityIntegralreal analysiscollege contestsfunction

Problem Statement

MM is a postive real and f:[0,∞)→[0,M]f:[0,\infty)\to[0,M] is a continuous function such that ∫0∞(1+x)f(x)dx<∞\int_0^\infty (1+x)f(x) dx<\infty Then, prove the following inequality. (∫0∞f(x)dx)2≤4M∫0∞xf(x)dx\left(\int_0^\infty f(x) dx \right)^2 \leq 4M \int_0^\infty x f(x) dx (@below, Thank you. I fixed.)
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