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Romania NMO 2023 Grade 12 P4

Source: Romania National Olympiad 2023

April 14, 2023
real analysisintegrationmonotone functionsdifferentiable function

Problem Statement

Let f:[0,1]→Rf:[0,1] \rightarrow \mathbb{R} a non-decreasing function, f∈C1,f \in C^1, for which f(0)=0.f(0) = 0. Let g:[0,1]→Rg:[0,1] \rightarrow \mathbb{R} a function defined by
g(x)=f(x)+(x−1)f′(x),∀x∈[0,1]. g(x) = f(x) + (x - 1) f'(x), \forall x \in [0,1].
a) Show that
∫01g(x)dx=0. \int_{0}^{1} g(x) \text{dx} = 0.
b) Prove that for all functions ϕ:[0,1]→[0,1],\phi :[0,1] \rightarrow [0,1], convex and differentiable with ϕ(0)=0\phi(0) = 0 and ϕ(1)=1,\phi(1) = 1, the inequality holds
∫01g(ϕ(t))dt≤0. \int_{0}^{1} g( \phi(t)) \text{dt} \leq 0.
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