A minimum with no derivative The function f(x)=|x| has an...

A minimum with no derivative The function $f(x)=|x|$ has an
absolute minimum value at $x=0$ even though $f$ is not differen-
tiable at $x=0 .$ Is this consistent with Theorem 2$?$ Give reasons
for your answer.

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Key Concepts

Necessary Conditions for Local Extrema

The theorem often referred to in this context states that if a function is differentiable at a local extremum, then its derivative must equal zero. However, the absence of differentiability does not contradict the existence of an extremum, since the theorem only applies when differentiability is established. Thus, a minimum occurring at a point where the function is not differentiable does not violate the theorem.

Differentiability

Differentiability concerns whether a function has a defined derivative at a given point. A function can have a minimum or maximum at a point even if it is not differentiable there, since differentiability is a condition needed to apply certain theorems but is not in itself necessary for an extremum to exist.

Critical Points

Critical points are points in the domain of a function where the derivative is zero or does not exist. They are prime candidates for local extrema, and a point failing to be differentiable can still be classified as a critical point if it is an extremum, as with functions that have 'sharp corners' or cusps.

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