maximums

In mathematics, the maximum of a set S is an element m in S such that every element of S is less than or equal to m. The maximum value is m. If such an element does not exist, S has no maximum. The concept extends to functions by considering the maximum value the function attains over its domain.

For a real-valued function f defined on a domain D, a global (or absolute) maximum occurs at x* in D if f(x) ≤ f(x*) for all x in D. The corresponding value f(x*) is the global maximum. A local (or relative) maximum at x* means there is a neighborhood around x* where f(x) ≤ f(x*). A point can be a local maximum without being a global maximum; for example, a constant function has every point as a local (and global) maximum.

Supremum: the least upper bound of the image of f on D. The supremum need not be attained; in that case there is no maximum, though the supremum still exists. If the domain is compact and f is continuous, a maximum exists (extreme value theorem). Finite sets always have a maximum. A continuous function on a closed interval [a,b] has a maximum.

In multivariable contexts, local maxima are often analyzed via the gradient and Hessian, with global maxima requiring the function values to be bounded above. Maximums are central to optimization, economics, statistics, and data analysis. The term maximums can refer to the set of points achieving the maximum, or to the maximum value itself.