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. This task is also important in Physics where the minima correspond to equilibria.