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Derivatives, existence of value

Source: ILL 1970 - Problem 25.

May 24, 2011
calculusderivativefunctioninequalitiesintegrationreal analysisreal analysis unsolved

Problem Statement

A real function ff is defined for 0≤x≤10\le x\le 1, with its first derivative f′f' defined for 0≤x≤10\le x\le 1 and its second derivative f′′f'' defined for 0<x<10<x<1. Prove that if f(0)=f′(0)=f′(1)=f(1)−1=0f(0)=f'(0)=f'(1)=f(1)-1 =0, then there exists a number 0<y<10<y<1 such that ∣f′′(y)∣≥4|f''(y)|\ge 4.
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