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prove that there exists \xi

Source: IMC 1998 day 1 problem 4

November 1, 2005
functioncalculusderivativereal analysisreal analysis unsolved

Problem Statement

The function f:R→Rf: \mathbb{R}\rightarrow\mathbb{R} is twice differentiable and satisfies f(0)=2,f′(0)=−2,f(1)=1f(0)=2,f'(0)=-2,f(1)=1. Prove that there is a ξ∈]0,1[\xi \in ]0,1[ for which we have f(ξ)⋅f′(ξ)+f′′(ξ)=0f(\xi)\cdot f'(\xi)+f''(\xi)=0.
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