Problem

Source: Romanian National Olympiad 2017, grade xi, p.4

Tags: function, calculus, derivative, MVT, real analysis, Darboux



Let be a function $ f $ of class $ \mathcal{C}^1[a,b] $ whose derivative is positive. Prove that there exists a real number $ c\in (a,b) $ such that $$ f(f(b))-f(f(a))=(f'(c))^2(b-a) . $$