Maksimiarvo

Maksimiarvo, in mathematical use commonly translated as “maximum value,” is the greatest value that a function or sequence attains over a given domain. It is the value that a function reaches at some point of its domain, and it is achieved by at least one input.

A related concept is the supremum (least upper bound). The supremum of a set of values may exist even if the function does not actually attain that value within the domain. Thus a maximum requires attainment at some input, whereas the supremum may be approached but not reached.

For a function f defined on a domain D, the maximum value M is present if there exists x0 in D such that f(x0) = M and f(x) ≤ M for all x in D. If such an x0 exists, the maximum is attained at x0. The extreme value theorem gives a sufficient condition: if f is continuous on a closed and bounded set (a compact domain), then f attains a maximum on that domain.

Examples help illustrate the distinction. The function f(x) = -x^2 + 4 on the real line has a maximum value of 4 attained at x = 0. On the open interval (0,1], the same function has a maximum value of 4 attained at x = 0? Here, the maximum is not located inside the domain; a better example is f(x) = x on the open interval (0,1), which has supremum 1 but no maximum, since 1 is not attained. For a finite set, the maximum value is simply the largest element of the set.

In practice, Maksimiarvo is a core concept in optimization, analysis, and numerical methods, guiding problems where one seeks the largest possible outcome within given constraints.