Non-differentiability and the search for maxima and minima

Points of non-differentiability have an important consequence in the search for maxima and minima: at them Fermat’s condition does not apply, so they must be considered as separate candidates.

Remark — Implication for function study

At points of non-differentiability the Fermat condition ”$f'(x_0)=0$ implies an extremum” does not apply: there can be a local maximum or minimum without $f'$ vanishing (indeed, without $f'$ existing at all). Examples: $|x|$ has a minimum at $0$; $-\sqrt{|x|}$ has a maximum at $0$. When looking for maxima/minima, to the stationary candidates ($f'=0$) one therefore adds the points of non-differentiability interior to the domain, and the boundaries of the domain (Weierstrass’s theorem).

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Topics: Derivatives
Concepts: Maximum and minimum · Non-differentiable point · Stationary point
Skills: Reasoning by cases · Studying a function