Extrema are minima and maxima. The singular forms of these words are extremum, minimum, and maximum.

Extrema may be “global” or “local”. A global minimum of a function is the lowest value that ever achieves. If you imagine the function as a surface, then a global minimum is the lowest point on that surface. Formally, it is said that has a global minimum at if .

A local minimum of a function is a point which has less value than all points “next to” it. If you imagine the function as a surface, then a local minimum is the bottom of a “valley” or “bowl” in the surface somewhere. Formally, it is said that has a local minimum at if a neighborhood of such that , .

If you flip the signs above to , you get the definitions of global and local maxima.

A “strict local minima” or “strict local maxima” means that nearby points are strictly less than or strictly greater than the critical point, rather than or . For instance, a strict local minima at has a neighborhood such that , .

A saddle point is a critical point which is not a local extremum.

A related concept is plateau.

Finding minima or maxima is an important task which is part of the field of optimization.