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Two times derivable real function

Source: RMO 2008, 11th Grade, Problem 3

April 30, 2008
functionalgebradomainreal analysiscalculusintegrationanalytic geometry

Problem Statement

Let f:R→R f: \mathbb R \to \mathbb R be a function, two times derivable on R \mathbb R for which there exist c∈R c\in\mathbb R such that \frac { f(b)\minus{}f(a) }{b\minus{}a} \neq f'(c) , for all a≠b∈R a\neq b \in \mathbb R. Prove that f''(c)\equal{}0.
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