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Twice differentiable function

Source: IMC 2019 Day 1 P3

July 31, 2019
functionreal analysiscollege contests

Problem Statement

Let f:(−1,1)→Rf:(-1,1)\to \mathbb{R} be a twice differentiable function such that 2f’(x)+xf''(x)\geqslant 1   \text{ for } x\in (-1,1). Prove that ∫−11xf(x)dx⩾13.\int_{-1}^{1}xf(x)dx\geqslant \frac{1}{3}.
Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karim Rakhimov, Scuola Normale Superiore and National University of Uzbekistan
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