Part BCSIR NET December 2023

Differentiability, mean value theorems, Taylor, L'Hôpital

Let $f : R \to R$ be a differentiable function such that f and its derivative f′ have no common zeros in [0, 1]. Which one of the following statements is true?

A.f never vanishes in [0, 1].
B.f has at most finitely many zeros in [0, 1].
C.f has infinitely many zeros in [0, 1].
D.f(1/2) = 0.

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Related counterexample: Differentiable ⇒ continuously differentiable

Differentiability, mean value theorems, Taylor, L'Hôpital

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