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 28 April 2018, at 15:40 (UTC).

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.

- Mathematical Analysis
- Function
- Domain Of A Function
- Pierre De Fermat
- Adequality
- Set Theory
- Set
- Greatest And Least Elements
- Real Numbers
- Function
- Domain
- Metric Space
- Topological Space
- Continuous
- Compact
- Interval
- Real Numbers

- Maxima And Minima
- Ice Cream Freeze (Let's Chill)
- Gold Reserve Act
- Windex.php?title=Manuel Ure%C3%B1a&action=edit&redlink=1
- Incentivisation
- Diwan (title)
- Uranus
- Mahalwari
- %C4%8C%C3%ADho%C5%A1%C5%A5 Miracle
- Lew Hoad
- Cornwallis In India
- Occupation Of The Baltic States
- William M. Meredith
- Smithville, New York
- Coventry, New York
- Loomis Family Farm
- Commons.wikimedia.orgTemplate:Path Text SVGzh-tw
- Congo Peafowl
- Pavo Muticus
- Pavo Cristatus
- Laureano Barrau
- Juan Jos%C3%A9 Or%C3%A9
- Huis Honselaarsdijk
- Ryotwari
- Charter Act Of 1833
- Varanasi District
- Edward Harris (Irish Judge)
- Apothecaries%27 System
- 2007 FIFA U-17 World Cup
- Guilford, New York