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IMC 2016, Problem 1

Source: IMC 2016

July 27, 2016
IMCIMC 2016functionsfunctionreal analysiscollege contests

Problem Statement

Let f:[a,b]→Rf : \left[ a, b\right]\rightarrow\mathbb{R} be continuous on [a,b]\left[ a, b\right] and differentiable on (a,b)\left( a, b\right). Suppose that ff has infinitely many zeros, but there is no x∈(a,b)x\in \left( a, b\right) with f(x)=f′(x)=0f(x)=f'(x)=0. (a) Prove that f(a)f(b)=0f(a)f(b)=0. (b) Give an example of such a function on [0,1]\left[ 0, 1\right].
(Proposed by Alexandr Bolbot, Novosibirsk State University)
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