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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 (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 ...
, the maximum and minimum of a set are the greatest and least elements 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 \max(f(x)), and the value of the function at a minimum point is called the of the function. Symbolically, this can be written as follows: :x_0 \in X is a global maximum point of function f:X \to \R, if (\forall x \in X)\, f(x_0) \geq f(x). 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 (X, d_X) be a metric space and function f:X \to \R. Then x_0 \in X is a local maximum point of function f if (\exists \varepsilon > 0) such that (\forall x \in X)\, d_X(x, x_0)<\varepsilon \implies f(x_0)\geq f(x). 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 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 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, 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, 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,
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, given sufficient differentiability. For any function that is defined piecewise, 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 200 feet of fencing and is trying to maximize the square footage of a rectangular enclosure, where x is the length, y is the width, and xy is the area: : 2x+2y = 200 : 2y = 200-2x : \frac = \frac : y = 100 - x : xy=x(100-x) The derivative with respect to x is: :\begin \fracxy&=\fracx(100-x) \\ &=\frac \left(100x-x^2 \right) \\ &=100-2x \end Setting this equal to 0 :0=100-2x :2x=100 :x=50 reveals that x=50 is our only critical point. Now retrieve the endpoints by determining the interval to which x is restricted. Since width is positive, then x>0, and since that implies that Plug in critical point as well as endpoints 0 and into and the results are 2500, 0, and 0 respectively. Therefore, the greatest area attainable with a rectangle of 200 feet of fencing is


Functions of more than one variable

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 :f(x,y)= x^2+y^2(1-x)^3,\qquad x,y \in \R, 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 \max(S). 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 functions. 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 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 * Derivative test *
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 Minima
a
ConvergenceApplication of Maxima and Minima with sub pages of solved problems
* {{Calculus topics Calculus Mathematical analysis Mathematical optimization Superlatives