Ääriarvo

Ääriarvo is a Finnish term that translates to "extremum" or "extreme value" in English, primarily used in the context of mathematics and calculus. It refers to the maximum or minimum value of a function within a given interval or over its entire domain.

There are two main types of extrema: global (or absolute) extrema and local (or relative) extrema. A global maximum is the largest value the function attains across its entire domain, while a global minimum is the smallest. Local extrema are the highest or lowest points in a specific neighborhood of the function, not necessarily across the entire domain.

In calculus, extrema are often found by examining critical points. These are points where the function's derivative is either zero or undefined. Fermat's theorem states that if a function f has a local extremum at a point c and f'(c) exists, then f'(c) = 0. Therefore, potential locations for local extrema are among the points where the derivative is zero or does not exist.

To distinguish between maximum and minimum values at critical points, the first derivative test or the second derivative test can be employed. The first derivative test analyzes the sign changes of the derivative around a critical point, while the second derivative test examines the sign of the second derivative at the critical point.

For functions defined on a closed interval, the Extreme Value Theorem guarantees that a continuous function will attain both a global maximum and a global minimum on that interval. These extrema can occur either at the critical points within the interval or at the endpoints of the interval.

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