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Question:

  • Let $f$ be a differentiable function. If the point c is a critical number, then either it is a local maximum, or local minimum, or an inflection point. $T/F$ ?

My opinion:

  • If c is a critical point then f'(c)=0 or undefined. So it may local maximum and local minimum.

  • If f '(c)=∞ then c is inflection point at the same time and if f '(c)=0 it may inflection point again.

  • But i can't find instance disproves this thesis.

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What about

$$f(x)=\begin{cases} x^2\sin\left(\frac{1}{x}\right) & x \neq 0\\ 0 & x=0 \end{cases}$$

at $x=0$?

$f$ is differentiable on $\mathbb R$ but $0$ is not a minimum, not a maximum and not an inflexion point.

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  • $\begingroup$ Your nick and answer very compatible. I get it thanks. $\endgroup$ – nadirhan Jan 9 at 13:50
  • $\begingroup$ The SAME example the professor did during my Calculus I course. 1977 AD $\endgroup$ – Raffaele Jan 9 at 15:17
  • $\begingroup$ @Raffaele I imagine that the example is used in may Calculus courses! $f$ is an easy differentiable locally oscillating map! $\endgroup$ – mathcounterexamples.net Jan 9 at 16:16

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