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 in calculus states that if a function f(x) is continuous on a closed interval [a,b], then the function has both a maximum and minimum value on that interval. In other words, if we graph the function f(x) on the interval [a,b], we are guaranteed to find points where the function reaches its highest and lowest values.
To prove the Extreme Value Theorem, we need to consider two cases:
1. If the function has a maximum or minimum value at an endpoint (i.e. at x=a or x=b), then the theorem holds trivially.
2. If the function has neither a maximum nor a minimum value at an endpoint, then it must have a critical point where the derivative is equal to zero or undefined. By the Extreme Value Theorem, the function must reach a maximum or minimum value at this critical point.
In summary, the Extreme Value Theorem is a powerful tool in calculus that allows us to guarantee the existence of both maximum and minimum values for a function on a closed interval.
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