Computer Science I / Applications of Derivatives
Practice question · True or false

A point where f(x)=0f'(x) = 0 is always a maximum or a minimum.

Hints
  1. Consider f(x)=x3f(x) = x^{3} at zero.
  2. The derivative can touch zero without changing sign.
Show the answer

False

Why

False. f(x)=x3f(x) = x^{3} has f(0)=0f'(0) = 0 yet is increasing on both sides, an inflection, neither max nor min. The first derivative locates candidates; deciding which requires the sign change around the point, or the second derivative, and f=0f''= 0 leaves even that test inconclusive.

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