Extreme Values

• Assume $f$ is defined on an open interval $I$ containing $c.$ If $f(c)\le f(x)$ for all $x$ in $I,$ then $f(c)$ is the minimum of $f$ on $I.$ If $f(c)\ge f(x)$ for all $x$ in $I,$ then $f(c)$ is the maximum of $f$ on $I.$

• The minimum/maximum values are the extreme values, or extrema, of $f$ on $I.$

• Extreme Value Theorem: If $f$ is continuous on $[a,b],$ then $f$ has both a minimum and a maximum on $[a,b].$

• If $f(c)$ is an extremum on an open interval containing $c,$ then $f(c)$ is a relative extremum.

• Relative extrema occur only at critical numbers.

• Extrema on a closed interval occur either at critical numbers or endpoints of the interval.