As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum.

A real-valued function *f* defined on a domain *X* has a **global** (or **absolute**) **maximum point** at *x*^{∗} if *f*(*x*^{∗}) ≥ *f*(*x*) for all *x* in *X*. Similarly, the function has a **global** (or **absolute**) **minimum point** at *x*^{∗} if *f*(*x*^{∗}) ≤ *f*(*x*) for all *x* in *X*. The value of the function at a maximum point is called the **maximum value** of the function and the value of the function at a minimum point is called the **minimum value** of the function.

If the domain *X* is a metric space then *f* is said to have a **local** (or **relative**) **maximum point** at the point *x*^{∗} if there exists some *ε* > 0 such that *f*(*x*^{∗}) ≥ *f*(*x*) for all *x* in *X* within distance *ε* of *x*^{∗}. Similarly, the function has a **local minimum point** at *x*^{∗} if *f*(*x*^{∗}) ≤ *f*(*x*) for all *x* in *X* within distance *ε* of *x*^{∗}. A similar definition can be used when *X* is a topological space, since the definition just given can be rephrased in terms of neighbourhoods.

In both the global and local cases, the concept of a **strict** extremum can be defined. For example, *x*^{∗} is a **strict global maximum point** if, for all *x* in *X* with *x* ≠ *x*^{∗}, we have *f*(*x*^{∗}) > *f*(*x*), and *x*^{∗} is a **strict local maximum point** if there exists some *ε* > 0 such that, for all *x* in *X* within distance *ε* of *x*^{∗} with *x* ≠ *x*^{∗}, we have *f*(*x*^{∗}) > *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 real-valued function with a 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 of real numbers (see the graph above).

This page was last edited on 6 March 2018, at 19:00.

Reference: https://en.wikipedia.org/wiki/Maxima_and_minima under CC BY-SA license.

Reference: https://en.wikipedia.org/wiki/Maxima_and_minima under CC BY-SA license.

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