IMC 2017 Problem 2

Let $f:\mathbb R\to(0,\infty)$ be a differentiabe function, and suppose that there exists a constant $L>0$ such that $$|f'(x)-f'(y)|\leq L|x-y|$$ for all $x,y$ . Prove that $$(f'(x))^2<2Lf(x)$$ holds for all $x$ .