The Fundamental Theorem For Extreme Values: Maximum And Minimum Values Of Continuous Functions On Closed Intervals

Extreme Value Theorem

If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occur at critical points in the interval or at endpoints of the interval.

The Extreme Value Theorem is a fundamental result in calculus that pertains to the existence of extreme values, such as maximum and minimum values, for continuous functions on closed intervals. This theorem states that if a function f(x) is continuous on a closed interval [a,b], then it has both a maximum and a minimum value on the interval. That is, there exist points c and d in the interval [a,b] such that f(c) is the maximum value of f(x) on the interval and f(d) is the minimum value of f(x) on the interval.

There are a few key points to take away from the Extreme Value Theorem. Firstly, it is important to note that continuity is a necessary condition for the existence of extreme values. This means that if a function is not continuous on a closed interval, then it may not have a maximum or minimum value on that interval.

Secondly, it is important to understand that the Extreme Value Theorem only applies to closed intervals. If an interval is open or half-open, then it may not have a maximum or minimum value. For example, the function f(x) = x^2 does not have a maximum value on the open interval (0,1), even though it has a maximum value on the closed interval [0,1].

Lastly, it is worth noting that the Extreme Value Theorem does not provide any information about where the maximum or minimum values occur on the interval. It only guarantees their existence. Therefore, it is often necessary to use additional techniques, such as derivative analysis, to determine the exact location of these extreme values.

In summary, the Extreme Value Theorem is a powerful tool for analyzing the behavior of continuous functions on closed intervals. It states that every such function has a maximum and minimum value on the interval, and understanding this theorem is essential for many applications in mathematics, economics, and other fields.

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