Summary of Maxima and Minima

A function may have both an absolute maximum and an absolute minimum, have just one absolute extremum, or have no absolute maximum or absolute minimum.
If a function has a local extremum, the point at which it occurs must be a critical point. However, a function need not have a local extremum at a critical point.
A continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. Each extremum occurs at a critical point or an endpoint.

Glossary

absolute extremum: if $f$ has an absolute maximum or absolute minimum at $c$ , we say $f$ has an absolute extremum at $c$

absolute maximum: if $f(c)\ge f(x)$ for all $x$ in the domain of $f$ , we say $f$ has an absolute maximum at $c$

absolute minimum: if $f(c)\le f(x)$ for all $x$ in the domain of $f$ , we say $f$ has an absolute minimum at $c$

critical point: if $f^{\prime}(c)=0$ or $f^{\prime}(c)$ is undefined, we say that $c$ is a critical point of $f$

extreme value theorem: if $f$ is a continuous function over a finite, closed interval, then $f$ has an absolute maximum and an absolute minimum

Fermat’s theorem: if $f$ has a local extremum at $c$ , then $c$ is a critical point of $f$

local extremum: if $f$ has a local maximum or local minimum at $c$ , we say $f$ has a local extremum at $c$

local maximum: if there exists an interval $I$ such that $f(c)\ge f(x)$ for all $x\in I$ , we say $f$ has a local maximum at $c$

local minimum: if there exists an interval $I$ such that $f(c)\le f(x)$ for all $x\in I$ , we say $f$ has a local minimum at $c$