Mathegoras

Calculus › Extrema and curve sketching › Bounds from derivatives

Smallest bound on function differences

High school · ●●●○○ 50 pts

The function f(x)f(x) is defined on [0,1][0,1] and satisfies f(0)=f(1)f(0)=f(1), and for any x,y∈[0,1]x, y \in [0,1] with x≠yx \neq y, we have ∣f(x)−f(y)∣<∣x−y∣|f(x)-f(y)|<|x-y|. Find the smallest real number mm such that for every such function ff and any x,y∈[0,1]x, y \in [0,1], we have ∣f(x)−f(y)∣<m|f(x)-f(y)|<m. Give your answer as a decimal.

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functional inequalitiessupremumoptimizationpiecewise functions