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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, maxima and minima, known collectively as extrema, are the largest value (maximum) or smallest value (minimum), that a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 takes in a point either within a given neighbourhood (local extremum) or on the function domain
Domain (mathematics)

In mathematics, the domain of a given function is the set of "input" values for which the function is defined. For instance, the domain of cosine would be all real numbers, while the domain of the square root would be only numbers greater than or equal to 0 ....
 in its entirety (global extremum).

Definitions
A real-valued function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 f defined on the real line
Real line

In mathematics, the real line is simply the set R of singleton real numbers.However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space....
 is said to have a local maximum point at the point x∗, if there exists some ε > 0, such that f(x∗) ≥ f(x) when |x - x∗| < ε.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, maxima and minima, known collectively as extrema, are the largest value (maximum) or smallest value (minimum), that a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 takes in a point either within a given neighbourhood (local extremum) or on the function domain
Domain (mathematics)

In mathematics, the domain of a given function is the set of "input" values for which the function is defined. For instance, the domain of cosine would be all real numbers, while the domain of the square root would be only numbers greater than or equal to 0 ....
 in its entirety (global extremum).

Definitions


A real-valued function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 f defined on the real line
Real line

In mathematics, the real line is simply the set R of singleton real numbers.However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space....
 is said to have a local maximum point at the point x∗, if there exists some ε > 0, such that f(x∗) ≥ f(x) when |x - x∗| < ε. The value of the function at this point is called maximum of the function.

On a graph
Graph of a function

In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian coordinate system, together with Cartesian axes, etc....
 of a function, its local maxima will look like the tops of hills.

Similarly, a function has a local minimum point at x∗, if f(x∗) ≤ f(x) when |x - x∗| < ε. The value of the function at this point is called minimum of the function.

On a graph of a function, its local minima will look like the bottoms of valleys.

A function has a global (or absolute) maximum point at x∗, if f(x∗) ≥ f(x) for all x.

Similarly, a function has a global (or absolute) minimum point at x∗, if f(x∗) ≤ f(x) for all x.

Any global maximum (minimum) point is also a local maximum (minimum) point; however, a local maximum or minimum point need not also be a global maximum or minimum point.

Terminology: The terms local and global are synonymous with relative and absolute respectively. Also extremum is an inclusive term that includes both maximum and minimum: a local extremum is a local or relative maximum or minimum, and a global extremum is a global or absolute maximum or minimum.

Restricted domains: There may be maxima and minima for a function whose domain
Domain (mathematics)

In mathematics, the domain of a given function is the set of "input" values for which the function is defined. For instance, the domain of cosine would be all real numbers, while the domain of the square root would be only numbers greater than or equal to 0 ....
 does not include all real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s. A real-valued function, whose domain is any set, can have a global maximum and minimum. There may also be local maxima and local minima points, but only at points of the domain set where the concept of neighborhood
Topology glossary

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology....
 is defined. A neighborhood plays the role of the set of x such that |x - x∗| < ε.

A continuous
Continuous function

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous....
 (real-valued) function on a compact
Compact space

In mathematics, a topological space is called compact if each of its open covers has a finite set subcover.Note: Some authors such as Nicolas Bourbaki use the term "quasi-compact" for this instead, and reserve the term "compact" for topological spaces that are both Hausdorff spaces and "quasi-compact"....
 set always takes maximum and minimum values on that set. An important example is a function whose domain is a closed (and bounded) interval
Interval

Interval may refer to:* Interval , a range of numbers * Interval measurements or interval variables in statistics is a level of measurement* Interval , the relationship between two notes...
 of real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s (see the graph above). The neighborhood
Topology glossary

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology....
 requirement precludes a local maximum or minimum at an endpoint of an interval. However, an endpoint may still be a global maximum or minimum. Thus it is not always true, for finite domains, that a global maximum (minimum) must also be a local maximum (minimum).

Terminology: The term optimum can replace either one of the terms maximum or minimum, depending on the context. Some optimization problems (see next paragraph) search for a global maximum value while others search for a global minimum value.

Finding maxima and minima


Finding global maxima and minima is the goal of optimization
Optimization (mathematics)

In mathematics, the simplest case of optimization, or mathematical programming, refers to the study of problems in which one seeks to maxima and minima or maxima and minima a Function of a real variable by systematically choosing the values of Real number or integer variables from within an allowed set....
. If a function is continuous on a closed interval, then by the extreme value theorem
Extreme value theorem

In calculus, the extreme value theorem states that if a real-valued Function f is Continuous function in the closed set and bounded interval [a,b], then f must attain its maximum and minimum value, each at least once....
 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 biggest (or smallest) one.

Local extrema can be found by Fermat's theorem
Fermat's theorem (stationary points)

Fermat's theorem is a theorem in real analysis, named after Pierre de Fermat. It gives a method to find local maximum and minimum of differentiable functions by showing that every local extremum of the function is a stationary point ....
, which states that they must occur at critical point
Critical point (mathematics)

In mathematics, a critical point is a Point on the domain of a function of a function where:* one dimension: the derivative is equality to 0 or a point where the function ceases to be differentiable....
s. One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test
First derivative test

In calculus, the first derivative test determines whether a given stationary point of a function is a maximum, a minimum, or neither....
 or second derivative test
Second derivative test

In calculus, a branch of mathematics, the second derivative test is a criterion often useful for determining whether a given stationary point of a function is a local Maxima and minima or a local Maxima and minima....
.

For any function that is defined piecewise
Piecewise

In mathematics, a piecewise-defined function is a function whose definition is dependent on the value of the independent variable. Mathematically, a real number-valued function f of a real variable x is a relationship whose definition is given differently on disjoint subsets of its domain ....
, one finds maxima (or minima) by finding the maximum (or minimum) of each piece separately; and then seeing which one is biggest (or smallest).

Examples


  • The function x2 has a unique global minimum at x = 0.
  • The function x3 has no global or local minima or maxima. Although the first derivative (3x2) is 0 at x = 0, this is an inflection point
    Inflection point

    In differential calculus, an inflection point, or point of inflection is a point on a curve at which the curvature changes Negative and non-negative numbers....
    .
  • The function x3/3 - x has first derivative x2 - 1 and second derivative 2x. Setting the first derivative to 0 and solving for x gives stationary points at -1 and +1. From the sign of the second derivative we can see that -1 is a local maximum and +1 is a local minimum. Note that this function has no global maximum or minimum.
  • The function |x| has a global minimum at x = 0 that cannot be found by taking derivatives, because the derivative does not exist at x = 0.
  • The function cos(x) has infinitely many global maxima at 0, ±2π, ±4π, …, and infinitely many global minima at ±π, ±3π, ….
  • The function 2 cos(x) - x has infinitely many local maxima and minima, but no global maximum or minimum.
  • The function cos(3πx)/x with 0.1 ≤ x ≤ 1.1 has a global maximum at x = 0.1 (a boundary), a global minimum near x = 0.3, a local maximum near x = 0.6, and a local minimum near x = 1.0. (See figure at top of page.)
  • The function x3 + 3x2 - 2x + 1 defined over the closed interval (segment) [-4,2] has two extrema: one local maximum at x = -1-√15⁄3, one local minimum at x = -1+√15⁄3, a global maximum at x = 2 and a global minimum at x = -4. (See figure at right)


Functions of more variables

For functions of more than one variable, similar conditions apply.

For example, in the (enlargeable) figure at the right, the necessary conditions for a local maximum are similar to those of a function with only one variable. The first partial derivatives 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
Saddle point

In mathematics, a saddle point is a point in the domain of a function of two variables which is a stationary point but not a local extremum....
. For use of these conditions to solve for a maximum, the function z must also be differentiable throughout. The second partial derivative test
Second partial derivative test

In mathematics, the second partial derivatives test is a method in multivariable calculus used to determine if a critical point is a minimum, maximum or saddle point....
 can help classify the point as a relative maximum or relative minimum.

A counterexample

However, for identifying global maxima and minima, there are substantial differences between functions of one and several variables. For example, if a differentiable function f defined on 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
Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the of a continuous function there is a corresponding value in its domain mapping to the original....
 and Rolle's theorem
Rolle's theorem

In calculus, a branch of mathematics, Rolle's theorem essentially states that a differentiable function , which attains equal values at two points, must have a stationary point somewhere between them where the slope is zero....
 to prove this by reductio ad absurdum
Reductio ad absurdum

Reductio ad absurdum , also known as an apagogical argument, reductio ad impossibile, or proof by contradiction, is a type of logical argument where one assumes a claim for the sake of argument and derives an absurd or ridiculous outcome, and then concludes that the original claim must have been wrong as it led to an abs...
). In two and more dimensions, this argument fails, as the function shows. Its 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(4,1) = −11.

See also

  • First derivative test
    First derivative test

    In calculus, the first derivative test determines whether a given stationary point of a function is a maximum, a minimum, or neither....
  • Second derivative test
    Second derivative test

    In calculus, a branch of mathematics, the second derivative test is a criterion often useful for determining whether a given stationary point of a function is a local Maxima and minima or a local Maxima and minima....
  • Limit superior and limit inferior
    Limit superior and limit inferior

    In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence. The limit inferior and limit superior of a function can be thought of in a similar fashion The limit inferior and limit superior of a set are the infimum and supremum of the set's limit points respectively....
  • Mechanical equilibrium
    Mechanical equilibrium

    A standard definition of is:This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next....
  • Extreme value
    Extreme value

    The largest and the smallest element of a Set are called extreme values, absolute extrema, or extreme records.For a differentiable function , if is an extreme value for the set of all values , and if is in the interior of the domain of , then is a Critical_point_, by Fermat's theorem ....


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