# extremum

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

Extrema may be “global” or “local”. A global minimum of a function $f$ is the lowest value that $f$ 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 $f\colon U\to V$ has a global minimum at $x$ if $\forall u\in U,f(x)\leq f(u)$.

A local minimum of a function $f$ is a point $x$ which has less value than all points “next to” it. If you imagine the function as a surface, then a local minimum is the of a “valley” or “bowl” in the surface somewhere. Formally, it is said that $f\colon U\to V$ has a local minimum at $x$ if $\exists$ a neighborhood $N$ of $x$ such that $\forall y\in N$, $f(x)\leq f(y)$.

If you flip the $\leq$ signs above to $\geq$, 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 $\leq$ or $\geq$. For instance, a strict local minima at $x$ has a neighborhood $N$ such that $\forall y\in N$, $(f(x).

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

 Title extremum Canonical name Extremum Date of creation 2013-03-22 12:59:44 Last modified on 2013-03-22 12:59:44 Owner bshanks (153) Last modified by bshanks (153) Numerical id 20 Author bshanks (153) Entry type Definition Classification msc 26B12 Synonym extrema Related topic Plateau Related topic RelationsBetweenHessianMatrixAndLocalExtrema Related topic LeastAndGreatestValueOfFunction Related topic SpeediestInclinedPlane Defines global minima Defines global maxima Defines local minima Defines local maxima Defines global minimum Defines global maximum Defines local minimum Defines local maximum Defines strict local minima Defines strict local maxima Defines strict local minimum Defines strict local maximum Defines saddle point