Maxima and minima

In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, 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).[1][2][3] Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.

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.

Local and global maxima and minima for cos(3πx)/x, 0.1≤ x ≤1.1

1 Definition

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, denoted {\displaystyle \max(f(x))}{\displaystyle \max(f(x))}, and the value of the function at a minimum point is called the minimum value of the function. Symbolically, this can be written as follows:

{\displaystyle x_{0}\in X}{\displaystyle x_{0}\in X} is a global maximum point of function {\displaystyle f:X\to \mathbb {R} ,}{\displaystyle f:X\to \mathbb {R} ,} if {\displaystyle (\forall x\in X),f(x_{0})\geq f(x).}{\displaystyle (\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, 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. Mathematically, the given definition is written as follows:

Let {\displaystyle (X,d_{X})}{\displaystyle (X,d_{X})} be a metric space and function {\displaystyle f:X\to \mathbb {R} }{\displaystyle f:X\to \mathbb {R} }. Then {\displaystyle x_{0}\in X}{\displaystyle x_{0}\in X} is a local maximum point of function {\displaystyle f}f if {\displaystyle (\exists \varepsilon >0)}{\displaystyle (\exists \varepsilon >0)} such that {\displaystyle (\forall x\in X),d_{X}(x,x_{0})<\varepsilon \implies f(x_{0})\geq f(x).}{\displaystyle (\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 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).

2 Search

3 Examples

4 Functions of more than one variable

5 Maxima or minima of a functional

6 In relation to sets

7 See also