äärimmäisyysarvon

Äärimmäisyysarvo, often translated as extremum value or extreme value in English, is a concept used in mathematics, particularly in calculus and optimization, to refer to the maximum or minimum value of a function. These values can occur either within a given interval (local or relative extremum) or over the entire domain of the function (global or absolute extremum).

A local maximum is a point where the function's value is greater than or equal to the values at all nearby points. Conversely, a local minimum is a point where the function's value is less than or equal to the values at all nearby points. For differentiable functions, local extrema often occur at critical points, which are points where the first derivative is either zero or undefined.

A global maximum is the largest value the function attains over its entire domain, and a global minimum is the smallest value. Global extrema are particularly important in applied problems where one seeks to maximize profit, minimize cost, or find the most efficient solution. The Extreme Value Theorem guarantees that a continuous function on a closed interval will attain both a global maximum and a global minimum within that interval. Identifying and calculating these äärimmäisyysarvot is a fundamental task in many areas of mathematics and its applications.