Understanding the Extreme Value Theorem in Calculus | Finding Maximum and Minimum Values on a Closed Interval

extreme value theorem

The extreme value theorem is a fundamental result in calculus that relates to the existence of maximum and minimum values of a function on a closed interval

The extreme value theorem is a fundamental result in calculus that relates to the existence of maximum and minimum values of a function on a closed interval.

Formally, the extreme value theorem states that if a function f(x) is continuous on a closed interval [a, b], then f(x) attains its maximum and minimum values at least once on that interval. In other words, there exist numbers c and d in the interval [a, b] such that f(c) is the maximum value of f(x) on [a, b] and f(d) is the minimum value of f(x) on [a, b].

This theorem ensures that continuous functions defined on a closed interval will always have a highest and lowest point within that interval. It is important to note that the maximum and minimum values may occur at the endpoints of the interval or at interior points within the interval.

The extreme value theorem is a powerful tool in calculus as it allows us to determine the maximum and minimum values of functions without needing to check every individual point. It provides a guarantee that the maximum and minimum values exist as long as the function is continuous on the given interval.

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