MathDB

8

Part of 2023 ISI Entrance UGB

Problems(1)

Inequality in MVT?

Source: ISI Entrance UGB 2023/8

5/14/2023
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)|.
LMVTMVTcontinuitydifferentiabilityfunctioncalculus