Extreme Value TheoremIf f is continuous on the closed interval [a, b] then…
f has a maximum and minimum value on the interval [a, b]
The Extreme Value Theorem states that a continuous function f defined on a closed interval [a, b] will have both a maximum and a minimum value on that interval. In other words, there are two points c and d in the interval [a, b] such that f(c) is the maximum value and f(d) is the minimum value of the function f on the interval [a, b].
This theorem is important because it assures us that if a function is continuous on a closed interval, we can always find the largest and smallest values of that function on that interval, which can often have practical applications.
It is worth noting that the theorem only holds true for continuous functions on closed intervals. If the function is not continuous or the interval is not closed, then the maximum or minimum value may not exist.
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