enimmäiset

Enimmäiset is the Finnish term for maximum values, referring to the greatest values that a function, sequence, or data set attains on its domain. In mathematics, they are called maxima, with distinctions between global (or absolute) maxima and local (or relative) maxima.

A global maximum of a function f on a domain D is a point x0 in D such that f(x0) is greater than or equal to f(x) for all x in D. A local maximum is a point x1 in D where f(x1) is at least as large as f(x) for all x in some small neighborhood around x1. Maxima can be strict (the value is greater than nearby points) or non-strict (ties with nearby points). The concept also applies to sequences, where a term is a maximum if no later term exceeds it within the considered index set.

Existence is not guaranteed. A function can be unbounded on its domain and thus have no maximum, even if a supremum (least upper bound) exists. By the Weierstrass theorem, a continuous function on a compact (closed and bounded) domain attains both a maximum and a minimum, ensuring enimmäiset in that setting.

Examples help illustrate the ideas: f(x) = -x^2 on the real numbers has a global maximum at x = 0 with value 0. The constant function f(x) = 1 has a global maximum of 1 everywhere. The function f(x) = x on R has no maximum because its values grow without bound.

In multiple variables and constrained problems, maxima are found using methods such as calculus of several variables or Lagrange multipliers. In statistics and data analysis, the term may also describe the largest observed value within a dataset.