IMC 2017 · Problem 2

Let $f \colon \mathbb{R} \to (0, \infty)$ be a differentiable function, and suppose that there exists a constant $L > 0$ such that

$$|f'(x) - f'(y)| \le L|x - y|$$

for all $x, y$. Prove that

$$\left(f'(x)\right)^2 < 2Lf(x)$$

holds for all $x$.