Äärimmäisarvolause

Äärimmäisarvolause is a mathematical theorem that deals with continuous functions on closed intervals. It states that if a function f is continuous on a closed interval [a, b], then f must attain both an absolute maximum value and an absolute minimum value on that interval. This means there exist numbers c and d in the interval [a, b] such that f(c) >= f(x) for all x in [a, b] (absolute maximum) and f(d) <= f(x) for all x in [a, b] (absolute minimum).

The theorem is fundamental in calculus and analysis. The conditions of continuity and the interval being closed are crucial. A function that is not continuous on a closed interval might not have an absolute maximum or minimum. Similarly, a continuous function on an open interval may not attain its extreme values. For example, the function f(x) = 1/x is continuous on the open interval (0, 1), but it does not have an absolute maximum or minimum on this interval.

The proof of the äärimmäisarvolause typically relies on the Bolzano-Weierstrass theorem, which guarantees the existence of convergent subsequences for bounded sequences. The theorem guarantees the existence of these extreme values, but it does not provide a method for finding them. Finding the actual maximum and minimum values usually involves examining critical points (where the derivative is zero or undefined) and the endpoints of the interval.