Extreme Value Theorem
If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b].
The Extreme Value Theorem is a fundamental theorem in calculus that states that if a real-valued function f is continuous on a closed interval [a, b], then f has a maximum and a minimum value on that interval. In other words, if we consider all the possible values of f(x) for x in [a, b], then we can find the largest and smallest values of f, and these values can be achieved at some point(s) within the interval.
This theorem is important in calculus because it allows us to determine the maximum and minimum values of a function on a given interval without having to check every point on the interval. Instead, we can just evaluate the function at the endpoints of the interval and any critical points within the interval (where the derivative of the function is zero or undefined), and the maximum and minimum values will be among these values.
For example, if we have a continuous function f(x) on the interval [-2, 2], we can use the Extreme Value Theorem to determine the maximum and minimum values of the function on that interval. We would evaluate the function at x = -2, x = 2, and any critical points within the interval. The largest value obtained would be the maximum value of the function on the interval, and the smallest value obtained would be the minimum value of the function on the interval.
Overall, the Extreme Value Theorem is a powerful tool in calculus that allows us to quickly and efficiently find the maximum and minimum values of a continuous function on a closed interval.
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