IMC 2005 · Problem 3

Let $f : \mathbb{R} \to [0,\infty)$ be a continuously differentiable function. Prove that

$$\left| \int_0^1 f^3(x)\,dx - f^2(0) \int_0^1 f(x)\,dx \right| \le \max_{0 \le x \le 1} |f'(x)| \left( \int_0^1 f(x)\,dx \right)^2.$$