Extreme Value Theorem
The Extreme Value Theorem is a fundamental result in calculus that guarantees the existence of maximum and minimum values for a continuous function on a closed interval
The Extreme Value Theorem is a fundamental result in calculus that guarantees the existence of maximum and minimum values for a continuous function on a closed interval.
Formally stated, if a function f is continuous on a closed interval [a, b], then f has both a maximum value M and a minimum value m on that interval. Moreover, there exist points c and d in the interval [a, b] such that f(c) = M and f(d) = m.
The Extreme Value Theorem is significant because it assures us that for continuous functions defined on closed intervals, extreme values are attainable. This means that if we want to find the absolute maximum or minimum values of a continuous function on a closed interval, we know that they exist and can be found through either careful examination of the function or by using calculus techniques such as finding critical points and endpoints.
To illustrate this, let’s consider an example. Suppose we have a function f(x) = x^2 on the closed interval [-1, 2]. By the Extreme Value Theorem, we know that this function has both a maximum and minimum value on the interval [-1, 2]. By observing the function or finding its derivative, we can see that the minimum value occurs at x = 0 (where f(0) = 0) and the maximum value occurs at x = 2 (where f(2) = 4). The Extreme Value Theorem guarantees that these extreme values exist and can be identified.
In summary, the Extreme Value Theorem is a powerful mathematical result that establishes the existence of maximum and minimum values for continuous functions on closed intervals. It provides assurance that when dealing with such functions, we can always find the highest and lowest points within a given interval.
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