IMC 2019 Problem 3

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