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Part BCSIR NET December 2023

Differentiability, mean value theorems, Taylor, L'Hôpital

Let f:R→Rf : \mathbb{R} \to \mathbb{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?

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

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

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Related counterexample: Differentiable ⇒ continuously differentiable

More on this topic

The chapter behind this: Differentiation — MVT, Darboux, Taylor and the classic spoilers — free to read

From Continuity and Differentiation › Differentiability, mean value theorems, Taylor, L'Hôpital

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