IMC 2019 · Day 1 · P3

Hint: $2 f'(x) + x f''(x)$ is the second derivative of a certain function.

Let $$g(x) = x f(x) - \frac{x^2}{2}.$$ Notice that $$g''(x) = 2 f'(x) + x f''(x) - 1 \ge 0,$$ so $g$ is convex. Estimate $g$ by its tangent at 0: let $g'(0) = a$, then $$g(x) = g(0) + g'(0) x = ax$$ and therefore $$\int_{-1}^{1} x f(x)\,dx = \int_{-1}^{1} \Bigl( g(x) + \frac{x^2}{2} \Bigr)\,dx \ge \int_{-1}^{1} \Bigl( ax + \frac{x^2}{2} \Bigr)\,dx = \frac13.$$