MathDB
A sufficient condition for nondecreasing monotony; examples

Source: Romanian District Olympiad 2016, Grade XI, Problem 4

October 5, 2018
functionanalysisDarbouxreal analysis

Problem Statement

Let I I be an open real interval, and let be two functions f,g:I⟶R f,g:I\longrightarrow\mathbb{R} satisfying the identity: x,y∈I∧x≠y  ⟹  f(x)−g(y)x−y+∣x−y∣≥0. x,y\in I\wedge x\neq y\implies\frac{f(x)-g(y)}{x-y} +|x-y|\ge 0.
a) Prove that f,g f,g are nondecreasing. b) Give a concrete example for f≠g. f\neq g.
Back to ProblemsView on AoPS