Essential Concepts
- 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 [latex]f[/latex] has an absolute maximum or absolute minimum at [latex]c[/latex], we say [latex]f[/latex] has an absolute extremum at [latex]c[/latex]
- absolute maximum
- if [latex]f(c)\ge f(x)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex], we say [latex]f[/latex] has an absolute maximum at [latex]c[/latex]
- absolute minimum
- if [latex]f(c)\le f(x)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex], we say [latex]f[/latex] has an absolute minimum at [latex]c[/latex]
- critical point
- if [latex]f^{\prime}(c)=0[/latex] or [latex]f^{\prime}(c)[/latex] is undefined, we say that [latex]c[/latex] is a critical point of [latex]f[/latex]
- extreme value theorem
- if [latex]f[/latex] is a continuous function over a finite, closed interval, then [latex]f[/latex] has an absolute maximum and an absolute minimum
- Fermat’s theorem
- if [latex]f[/latex] has a local extremum at [latex]c[/latex], then [latex]c[/latex] is a critical point of [latex]f[/latex]
- local extremum
- if [latex]f[/latex] has a local maximum or local minimum at [latex]c[/latex], we say [latex]f[/latex] has a local extremum at [latex]c[/latex]
- local maximum
- if there exists an interval [latex]I[/latex] such that [latex]f(c)\ge f(x)[/latex] for all [latex]x\in I[/latex], we say [latex]f[/latex] has a local maximum at [latex]c[/latex]
- local minimum
- if there exists an interval [latex]I[/latex] such that [latex]f(c)\le f(x)[/latex] for all [latex]x\in I[/latex], we say [latex]f[/latex] has a local minimum at [latex]c[/latex]
