Extreme Value Theorem
If f is continuous on a closed interval [a,b], then f has both a maximum value and a minimum value on the interval.
The Extreme Value Theorem states that if a function is continuous over a closed interval, then the function must have both a maximum and a minimum value on that interval. In other words, if we consider a function f(x) to be continuous on a closed interval, [a, b], then there exist points x = c and x = d within the interval such that f(c) is the maximum value of the function and f(d) is the minimum value of the function.
This theorem is essential in calculus and real analysis as it guarantees the existence of maximum and minimum values of a continuous function. We can use it to determine the maximum and minimum values of the function on a given interval. It is also useful in optimization problems, where we want to find the maximum or minimum value of a function subject to certain constraints.
To apply the Extreme Value Theorem, one must verify that the function is continuous over a closed interval. This means that the function must be defined and continuous at every point within the interval [a, b]. If the function is not continuous at any point within this interval, then the theorem does not apply.
Overall, the Extreme Value Theorem is a powerful tool in mathematics and is used in various fields, including economics, physics, and engineering.
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