Maksimit

Maksimit is a concept used in mathematics, statistics, and optimization to denote the greatest value that a quantity can attain within a defined context, such as a set, a function’s domain, or a data sample. It serves as a counterpart to the minimum and is central to analysis and decision making in quantitative disciplines.

The maxima of a quantity can be classified as global (absolute) maximum, which is the largest value over the entire domain, or local maximum, which is the largest value within a nearby neighborhood. A function may have several local maxima, and in some cases the maximum does not exist because the domain is unbounded or the function grows without bound.

Existence and identification of maxima are fundamental topics in calculus and analysis. For continuous functions on a closed and bounded interval, the extreme value theorem guarantees at least one maximum and one minimum. In optimization, constraints are handled with methods such as Lagrange multipliers, convex analysis, or numerical algorithms when analytical solutions are not feasible.

Practical computation uses analytical techniques, numerical optimization, or discrete search. For a discrete set, the maximum is found by comparing values. For a continuous function, one locates critical points by solving f′(x) = 0 and then tests these points, often with the second derivative test or first derivative test to determine whether a point yields a maximum.

Example: The maximum of the finite set {2, 7, 3, 9, 9, 4} is 9. For the function f(x) = −x^2 + 4x on the real line, the global maximum occurs at x = 2 with f(2) = 4. Related concepts include the local maximum, global maximum, and various optimization methods.