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IMC 1994 D2 P3

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March 6, 2017
IMCreal analysisfunctioncalculusderivative

Problem Statement

Let ff be a real-valued function with n+1n+1 derivatives at each point of R\mathbb R. Show that for each pair of real numbers aa, bb, a<ba<b, such that ln⁡(f(b)+f′(b)+⋯+f(n)(b)f(a)+f′(a)+⋯+f(n)(a))=b−a\ln\left( \frac{f(b)+f'(b)+\cdots + f^{(n)} (b)}{f(a)+f'(a)+\cdots + f^{(n)}(a)}\right)=b-a there is a number cc in the open interval (a,b)(a,b) for which f(n+1)(c)=f(c)f^{(n+1)}(c)=f(c)
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