Extreme Value Theoremf is cont. in the closed interval [a,b] then…
If a function f is continuous in the closed interval [a, b], then the Extreme Value Theorem states that f has both a maximum value and a minimum value within that interval
If a function f is continuous in the closed interval [a, b], then the Extreme Value Theorem states that f has both a maximum value and a minimum value within that interval.
More formally, the theorem states that if f is continuous in the closed interval [a, b], then there exist numbers c and d in [a, b] such that f(c) is the maximum value of f in [a, b] and f(d) is the minimum value of f in [a, b]. In other words, f(c) is greater than or equal to f(x) for all x in [a, b], and f(d) is less than or equal to f(x) for all x in [a, b].
This theorem is important in mathematical analysis as it guarantees the existence of maximum and minimum values for continuous functions. It allows us to determine the maximum and minimum points of a function within an interval, which can be useful in various applications such as optimization problems and finding critical points.
In practical terms, the Extreme Value Theorem means that if a function is continuous over a closed interval, we can be confident that it attains both its highest and lowest values within that interval.
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