Counterexample

Is this true?

$f'(x_{0}) = 0$ and f′ changes sign nowhere near $x_{0} \Rightarrow f$ is constant near $x_{0}$

No — it is false.

The counterexample

The object that kills this claim comes with Trap Analysis.

86 counterexamples are free — including most of the ones Part C leans on hardest.

Keep practicing

Converse assumed

The theorem runs one way. You used it in the other.

See real exam options built on this mistake Drill statements like this

differentiation

Go deeper· Explained in full in Differentiation — MVT, Darboux, Taylor and the classic spoilers· Asked in CSIR NET June 2025· From Continuity and Differentiation › Differentiability, mean value theorems, Taylor, L'Hôpital

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