IMC 2019 · Problem 3

Let $f \colon (-1, 1) \to \mathbb{R}$ be a twice differentiable function such that

$$2f'(x) + xf''(x) \ge 1 \quad \text{for } x \in (-1, 1).$$

Prove that

$$\int_{-1}^{1} xf(x)\,\mathrm{d}x \ge \frac{1}{3}.$$