In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extrema, they may be defined either within a given range (the local or relative extrema) or on the entire domain (the global or absolute extrema) of a function. 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. In statistics, the corresponding concept is the sample maximum and minimum.
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 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, (denoted min ( f ( x ) ) {\displaystyle \min(f(x))} for clarity). Symbolically, this can be written as follows:
x 0 ∈ X {\displaystyle x_{0}\in X} is a global maximum point of function f : X → R , {\displaystyle f:X\to \mathbb {R} ,} if ( ∀ x ∈ X ) f ( x 0 ) ≥ 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 ( X , d X ) {\displaystyle (X,d_{X})} be a metric space and function f : X → R {\displaystyle f:X\to \mathbb {R} } . Then x 0 ∈ X {\displaystyle x_{0}\in X} is a local maximum point of function f {\displaystyle f} if ( ∃ ε > 0 ) {\displaystyle (\exists \varepsilon >0)} such that ( ∀ x ∈ X ) d X ( x , x 0 ) < ε ⟹ f ( x 0 ) ≥ 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).
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 greatest (or least) 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 often distinguish whether a critical point is a local maximum, a local minimum, or neither by using the first derivative test, second derivative test, 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 greatest (or least).
Examples
… excerpt ends here. Continue reading the full article.
