### International Mathematics Competition for University Students

July 31 – August 6 2017, Blagoevgrad, Bulgaria

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2. Let $f\colon\mathbb{R}\to(0,\infty)$ be a differentiable function, and suppose that there exists a constant $L>0$ such that $$\bigl|f'(x)-f'(y)\bigr| \leq L\bigl|x-y\bigr|$$ for all $x,y$. Prove that $$\big(f'(x)\big)^2 < 2Lf(x)$$ holds for all $x$.
Hint: Integrate $f'$ over an interval $[x,x+\Delta]$.