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fg>=4id^2 implies int f or int g greater or equal than 1

Source: Romanian District Olympiad 2017, Grade XII, Problem 1

October 10, 2018
functionIntegralinequalitiescalculus

Problem Statement

Let f,g:[0,1]⟶R f,g:[0,1]\longrightarrow{R} be two continuous functions such that f(x)g(x)≥4x2, f(x)g(x)\ge 4x^2, for all x∈[0,1]. x\in [0,1] . Prove that ∣∫01f(x)dx∣≥1 or ∣∫01g(x)dx∣≥1. \left| \int_0^1 f(x)dx \right| \ge 1\text{ or } \left| \int_0^1 g(x)dx \right| \ge 1.
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