IMC 2025 · Problem 2

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a twice continuously differentiable function, and suppose that $\int_{-1}^{1} f(x)\,\mathrm{d}x = 0$ and $f(1) = f(-1) = 1$. Prove that

$$\int_{-1}^{1} \left(f''(x)\right)^2 \mathrm{d}x \ge 15,$$

and find all such functions for which equality holds.