ekstremaalväärtuseks

An ekstremaalväärtuseks, meaning "extreme value" in Norwegian, refers to the maximum or minimum value of a function within a specific interval or on its entire domain. These extreme values are crucial in calculus for understanding the behavior of functions.

There are two primary types of extreme values: absolute (or global) and relative (or local). An absolute maximum is the highest value the function attains across its entire domain, while an absolute minimum is the lowest. Relative maximums and minimums are the highest and lowest points within a particular neighborhood of the function, respectively. These points do not necessarily represent the absolute highest or lowest values.

To find extreme values, mathematicians use the concept of critical points. A critical point of a function is a point where the derivative is either zero or undefined. Fermat's theorem states that if a function f has a local extremum at a point c in an open interval (a, b), and if f'(c) exists, then f'(c) = 0. Therefore, potential locations for extreme values occur at critical points or at the endpoints of a closed interval.

The Extreme Value Theorem guarantees that a continuous function on a closed interval will always attain both an absolute maximum and an absolute minimum value within that interval. These extreme values are fundamental to optimization problems in various fields, including economics, engineering, and physics.