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'extremum'
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| Title of object: |
extremum |
| Canonical Name: |
Extremum |
| Type: |
Definition |
| Created on: |
2002-08-28 02:17:11 |
| Modified on: |
2008-09-22 08:15:44 |
| Classification: |
msc:26B12 |
| Defines: |
global minima, global maxima, local minima, local maxima, global minimum, global maximum, local minimum, local maximum, strict local minima, strict local maxima, strict local minimum, strict local maximum, saddle point |
| Synonyms: |
extremum=extrema |
Preamble:
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\usepackage{amssymb}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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Content:
Extrema are minima and maxima. The \PMlinkescapetext{singular} forms of these \PMlinkescapetext{words} 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 \emph{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 \PMlinkescapetext{bottom} of a ``valley'' or ``bowl'' in the surface somewhere. Formally, it is said that $f\colon U \to V$ has a \emph{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 $\forall y \in N$, $(f(x) < f(y) \textrm{ or } y = x)$.
A \emph{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 \PMlinkescapetext{field} of optimization.
This task is also important in Physics where the minima correspond to equilibria. |
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