Editing Maximum and minimum (section)

==Search==
Finding global maxima and minima is the goal of [[mathematical optimization]]. If a function is continuous on a closed interval, then by the [[extreme value theorem]], global maxima and minima exist. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary, and take the greatest (or least) one.

For [[differentiable functions]], [[Fermat's theorem (stationary points)|Fermat's theorem]] states that local extrema in the interior of a domain must occur at [[critical point (mathematics)|critical point]]s (or points where the derivative equals zero).<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Minimum|url=https://mathworld.wolfram.com/Minimum.html|access-date=2020-08-30|website=mathworld.wolfram.com|language=en}}</ref> However, not all critical points are extrema. One can often distinguish whether a critical point is a local maximum, a local minimum, or neither by using the [[first derivative test]], [[derivative test#Second-derivative test (single variable)|second derivative test]], or [[higher-order derivative test]], given sufficient differentiability.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Maximum|url=https://mathworld.wolfram.com/Maximum.html|access-date=2020-08-30|website=mathworld.wolfram.com|language=en}}</ref>

For any function that is defined [[piecewise]], one finds a maximum (or minimum) by finding the maximum (or minimum) of each piece separately, and then seeing which one is greatest (or least).