Necessary condition for an extremum

Differential calculus offers a powerful tool for locating the maximum and minimum points of a function: the first derivative. The first result links the presence of an extremum to the vanishing of the derivative.

Property — Necessary condition

If $x_0$ is a relative maximum or minimum point of $f$ and $f$ is differentiable at $x_0$, then $f'(x_0)=0$.

A point where the derivative vanishes is called a stationary point. The property therefore says that every “internal” differentiable relative extremum is necessarily a stationary point: to look for maxima and minima we can limit ourselves to examining the points where $f'(x)=0$.

Warning

The converse is false: $f'(x_0)=0$ does not imply that $x_0$ is an extremum. Example: $f(x)=x^3$ has $f'(0)=0$ but $x=0$ is an inflection point, not an extremum.

Finding the stationary points is therefore only the first step: a criterion is then needed to distinguish maxima, minima and inflections with a horizontal tangent.

Links

Topics: Function study
Concepts: First derivative · Relative maxima and minima · Stationary point
Methods: Maximum minimum via derivative
Skills: Differentiating