![Extrema example original](https://upload.wikimedia.org/wikipedia/commons/6/68/Extrema_example_original.svg)
In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, the maxima and minima (the respective plurals of maximum and minimum) of a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given
range
Range may refer to:
Geography
* Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra)
** Mountain range, a group of mountains bordered by lowlands
* Range, a term used to i ...
(the ''local'' or ''relative'' extrema), or on the entire
domain (the ''global'' or ''absolute'' extrema).
Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
was one of the first mathematicians to propose a general technique,
adequality Adequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam''[set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...](_blank)
, the maximum and minimum of a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
are the
greatest and least elements
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an ele ...
in the set, respectively. Unbounded
infinite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only s ...
s, such as the set of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, have no minimum or maximum.
Definition
A real-valued
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
''f'' defined on a
domain ''X'' has a global (or absolute) maximum point at ''x''
∗, if for all ''x'' in ''X''. Similarly, the function has a global (or absolute) minimum point at ''x''
∗, if for all ''x'' in ''X''. The value of the function at a maximum point is called the of the function, denoted
, and the value of the function at a minimum point is called the of the function. Symbolically, this can be written as follows:
:
is a global maximum point of function
if
The definition of global minimum point also proceeds similarly.
If the domain ''X'' is a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
, then ''f'' is said to have a local (or relative) maximum point at the point ''x''
∗, if there exists some ''ε'' > 0 such that 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
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
, since the definition just given can be rephrased in terms of neighbourhoods. Mathematically, the given definition is written as follows:
:Let
be a metric space and function
. Then
is a local maximum point of function
if
such that
The definition of local minimum point can also proceed similarly.
In both the global and local cases, the concept of a can be defined. For example, ''x''
∗ is a if for all ''x'' in ''X'' with , we have , and ''x''
∗ is a if there exists some such that, for all ''x'' in ''X'' within distance ''ε'' of ''x''
∗ with , we have . 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
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
real-valued function with a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
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 number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s (see the graph above).
Search
Finding global maxima and minima is the goal of
mathematical optimization. If a function is continuous on a closed interval, then by the
extreme value theorem
In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval ,b/math>, then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in ,b/math> su ...
, global maxima and minima exist. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary, and take the largest (or smallest) one.
For
differentiable functions
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in it ...
,
Fermat's theorem states that local extrema in the interior of a domain must occur at
critical points (or points where the derivative equals zero). However, not all critical points are extrema. One can distinguish whether a critical point is a local maximum or local minimum by using the
first derivative test
In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information abo ...
,
second derivative test
In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information abou ...
, or
higher-order derivative test
In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information abo ...
, given sufficient differentiability.
For any function that is defined
piecewise
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
, one finds a maximum (or minimum) by finding the maximum (or minimum) of each piece separately, and then seeing which one is largest (or smallest).
Examples
For a practical example,
assume a situation where someone has
feet of fencing and is trying to maximize the square footage of a rectangular enclosure, where
is the length,
is the width, and
is the area:
:
:
:
:
:
The derivative with respect to
is:
:
Setting this equal to
:
:
:
reveals that
is our only
critical point.
Now retrieve the
endpoints by determining the interval to which
is restricted. Since width is positive, then
, and since that implies that
Plug in critical point as well as endpoints
and into and the results are
and
respectively.
Therefore, the greatest area attainable with a rectangle of
feet of fencing is
Functions of more than one variable
![Modell einer Peanoschen Fläche -Schilling XLIX, 1-](https://upload.wikimedia.org/wikipedia/commons/2/23/Modell_einer_Peanoschen_Fl%C3%A4che_-Schilling_XLIX%2C_1-.jpg)
![MaximumCounterexample](https://upload.wikimedia.org/wikipedia/commons/9/95/MaximumCounterexample.png)
For functions of more than one variable, similar conditions apply. For example, in the (enlargeable) figure on the right, the necessary conditions for a ''local'' maximum are similar to those of a function with only one variable. The first
partial derivatives
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
as to ''z'' (the variable to be maximized) are zero at the maximum (the glowing dot on top in the figure). The second partial derivatives are negative. These are only necessary, not sufficient, conditions for a local maximum, because of the possibility of a
saddle point. For use of these conditions to solve for a maximum, the function ''z'' must also be
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
throughout. The
second partial derivative test can help classify the point as a relative maximum or relative minimum.
In contrast, there are substantial differences between functions of one variable and functions of more than one variable in the identification of global extrema. For example, if a bounded differentiable function ''f'' defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the
intermediate value theorem and
Rolle's theorem to prove this by
contradiction
In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
). In two and more dimensions, this argument fails. This is illustrated by the function
:
whose only critical point is at (0,0), which is a local minimum with ''f''(0,0) = 0. However, it cannot be a global one, because ''f''(2,3) = −5.
Maxima or minima of a functional
If the domain of a function for which an extremum is to be found consists itself of functions (i.e. if an extremum is to be found of a
functional), then the extremum is found using the
calculus of variations.
In relation to sets
Maxima and minima can also be defined for sets. In general, if an
ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
''S'' has a
greatest element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an elem ...
''m'', then ''m'' is a
maximal element of the set, also denoted as
. Furthermore, if ''S'' is a subset of an ordered set ''T'' and ''m'' is the greatest element of ''S'' with (respect to order induced by ''T''), then ''m'' is a
least upper bound
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
of ''S'' in ''T''. Similar results hold for
least element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an elem ...
,
minimal element
In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is defin ...
and
greatest lower bound. The maximum and minimum function for sets are used in
database
In computing, a database is an organized collection of data stored and accessed electronically. Small databases can be stored on a file system, while large databases are hosted on computer clusters or cloud storage. The design of databases s ...
s, and can be computed rapidly, since the maximum (or minimum) of a set can be computed from the maxima of a partition; formally, they are self-
decomposable aggregation function
In database management, an aggregate function or aggregation function is a function where the values of multiple rows are grouped together to form a single summary value.
Common aggregate functions include:
* Average (i.e., arithmetic mean)
* ...
s.
In the case of a general
partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
, the least element (i.e., one that is smaller than all others) should not be confused with a minimal element (nothing is smaller). Likewise, a
greatest element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an elem ...
of a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
(poset) is an
upper bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of .
Dually, a lower bound or minorant of is defined to be an eleme ...
of the set which is contained within the set, whereas a maximal element ''m'' of a poset ''A'' is an element of ''A'' such that if ''m'' ≤ ''b'' (for any ''b'' in ''A''), then ''m'' = ''b''. Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal elements. If a poset has more than one maximal element, then these elements will not be mutually comparable.
In a
totally ordered
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive ...
set, or ''chain'', all elements are mutually comparable, so such a set can have at most one minimal element and at most one maximal element. Then, due to mutual comparability, the minimal element will also be the least element, and the maximal element will also be the greatest element. Thus in a totally ordered set, we can simply use the terms ''minimum'' and ''maximum''.
If a chain is finite, then it will always have a maximum and a minimum. If a chain is infinite, then it need not have a maximum or a minimum. For example, the set of
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
s has no maximum, though it has a minimum. If an infinite chain ''S'' is bounded, then the
closure ''Cl''(''S'') of the set occasionally has a minimum and a maximum, in which case they are called the greatest lower bound and the least upper bound of the set ''S'', respectively.
See also
*
Arg max
In mathematics, the arguments of the maxima (abbreviated arg max or argmax) are the points, or elements, of the domain of some function at which the function values are maximized.For clarity, we refer to the input (''x'') as ''points'' and t ...
*
Derivative test
In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information abo ...
*
Infimum and supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
*
Limit superior and limit inferior
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a ...
*
Mechanical equilibrium
In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero. By extension, a physical system made up of many parts is in mechanical equilibrium if the net force on each of its individual parts is zero ...
*
Mex (mathematics)
*
Sample maximum and minimum
*
Saddle point
References
External links
Thomas Simpson's work on Maxima and Minimaa
ConvergenceApplication of Maxima and Minima with sub pages of solved problems*
{{Calculus topics
Calculus
Mathematical analysis
Mathematical optimization
Superlatives