Problem

Source: Romania National Olympiad 2023

Tags: real analysis, integration, monotone functions, differentiable function



Let $f:[0,1] \rightarrow \mathbb{R}$ a non-decreasing function, $f \in C^1,$ for which $f(0) = 0.$ Let $g:[0,1] \rightarrow \mathbb{R}$ a function defined by \[ g(x) = f(x) + (x - 1) f'(x), \forall x \in [0,1]. \] a) Show that \[ \int_{0}^{1} g(x) \text{dx} = 0. \] b) Prove that for all functions $\phi :[0,1] \rightarrow [0,1],$ convex and differentiable with $\phi(0) = 0$ and $\phi(1) = 1,$ the inequality holds \[ \int_{0}^{1} g( \phi(t)) \text{dt} \leq 0. \]