Editing Maximum and minimum (section)

==Definition==
A real-valued [[Function (mathematics)|function]] ''f'' defined on a [[Domain of a function|domain]] ''X'' has a '''global''' (or '''absolute''') '''maximum point'''{{anchor|Global maximum point|Absolute maximum point|Maximum point}} at ''x''<sup>∗</sup>, if {{nowrap|''f''(''x''<sup>∗</sup>) ≥ ''f''(''x'')}} for all ''x'' in ''X''. Similarly, the function has a '''global''' (or '''absolute''') '''minimum point'''{{anchor|Global minimum point|Absolute minimum point|Minimum point}} at ''x''<sup>∗</sup>, if {{nowrap|''f''(''x''<sup>∗</sup>) ≤ ''f''(''x'')}} for all ''x'' in ''X''. The value of the function at a maximum point is called the '''{{visible anchor|maximum value}}''' of the function, denoted <math>\max(f(x))</math>, and the value of the function at a minimum point is called the '''{{visible anchor|minimum value}}''' of the function, (denoted <math>\min(f(x))</math> for clarity). Symbolically, this can be written as follows:
:<math>x_0 \in X</math> is a global maximum point of function <math>f:X \to \R,</math> if <math>(\forall x \in X)\, f(x_0) \geq f(x).</math>
The definition of global minimum point also proceeds similarly.

If the domain ''X'' is a [[metric space]], then ''f'' is said to have a '''local''' (or '''relative''') '''maximum point'''{{anchor|Local maximum point|Relative maximum point}} at the point ''x''<sup>∗</sup>, if there exists some ''ε'' > 0 such that {{nowrap|''f''(''x''<sup>∗</sup>) ≥ ''f''(''x'')}} for all ''x'' in ''X'' within distance ''ε'' of ''x''<sup>∗</sup>. Similarly, the function has a '''local minimum point'''{{anchor|Local minimum point|Relative minimum point}} at ''x''<sup>∗</sup>, if ''f''(''x''<sup>∗</sup>) ≤ ''f''(''x'') for all ''x'' in ''X'' within distance ''ε'' of ''x''<sup>∗</sup>. A similar definition can be used when ''X'' is a [[topological space]], since the definition just given can be rephrased in terms of neighbourhoods. Mathematically, the given definition is written as follows:
:Let <math>(X, d_X)</math> be a metric space and function <math> f:X \to \R</math>. Then <math>x_0 \in X</math> is a local maximum point of function <math>f</math> if <math> (\exists \varepsilon > 0)</math> such that <math>(\forall x \in X)\, d_X(x, x_0)<\varepsilon \implies f(x_0)\geq f(x).</math>

The definition of local minimum point can also proceed similarly.

In both the global and local cases, the concept of a '''{{visible anchor|strict extremum}}''' can be defined. For example, ''x''<sup>∗</sup> is a '''{{visible anchor|strict global maximum point}}''' if for all ''x'' in ''X'' with {{nowrap|''x'' ≠ ''x''<sup>∗</sup>}}, we have {{nowrap|''f''(''x''<sup>∗</sup>) > ''f''(''x'')}}, and ''x''<sup>∗</sup> is a '''{{visible anchor|strict local maximum point}}''' if there exists some {{nowrap|''ε'' > 0}} such that, for all ''x'' in ''X'' within distance ''ε'' of ''x''<sup>∗</sup> with {{nowrap|''x'' ≠ ''x''<sup>∗</sup>}}, we have {{nowrap|''f''(''x''<sup>∗</sup>) > ''f''(''x'')}}. Note that a point is a strict global maximum point if and only if it is the unique global maximum point, and similarly for minimum points.

A [[Continuous function|continuous]] real-valued function with a [[Compact space|compact]] domain always has a maximum point and a minimum point. An important example is a function whose domain is a closed and bounded [[Interval (mathematics)|interval]] of [[real number]]s (see the graph above).