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Cauchy-like inequality implies that function has a root

Source: VJIMC 2024, Category II, Problem 1

April 14, 2024
inequalitiesfunctionexponentialcalculus

Problem Statement

Suppose that f:[−1,1]→Rf:[-1,1] \to \mathbb{R} is continuous and satisfies (∫−11exf(x)dx)2≥(∫−11f(x)dx)(∫−11e2xf(x)dx).\left(\int_{-1}^1 e^xf(x) dx\right)^2 \ge \left(\int_{-1}^1 f(x) dx\right)\left(\int_{-1}^1 e^{2x}f(x) dx\right). Prove that there exists a point c∈(−1,1)c \in (-1,1) such that f(c)=0f(c)=0.
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