MathDB
Inequality in MVT?

Source: ISI Entrance UGB 2023/8

May 14, 2023
LMVTMVTcontinuitydifferentiabilityfunctioncalculus

Problem Statement

Let f ⁣:[0,1]→Rf \colon [0,1] \to \mathbb{R} be a continuous function which is differentiable on (0,1)(0,1). Prove that either f(x)=ax+bf(x) = ax + b for all x∈[0,1]x \in [0,1] for some constants a,b∈Ra,b \in \mathbb{R} or there exists t∈(0,1)t \in (0,1) such that ∣f(1)−f(0)∣<∣f′(t)∣|f(1) - f(0)| < |f'(t)|.
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