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int f(x)g(x) dx <= int f(x)g(1-x) dx

Source: 1980 Swedish Mathematical Competition p4

March 28, 2021
integrationanalysisinequalities

Problem Statement

The functions ff and gg are positive and continuous. ff is increasing and gg is decreasing. Show that ∫01f(x)g(x)dx≤∫01f(x)g(1−x)dx \int\limits_0^1 f(x)g(x) dx \leq \int\limits_0^1 f(x)g(1-x) dx
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