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Putnam 2018 A5

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December 2, 2018
PutnamPutnam 2018Taylor seriescollege contestsreal analysis

Problem Statement

Let f:R→Rf: \mathbb{R} \to \mathbb{R} be an infinitely differentiable function satisfying f(0)=0f(0) = 0, f(1)=1f(1) = 1, and f(x)≥0f(x) \ge 0 for all x∈Rx \in \mathbb{R}. Show that there exist a positive integer nn and a real number xx such that f(n)(x)<0f^{(n)}(x) < 0.
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