Let h be a continuous function on the closed interval [0,4], where h(0)=2 and h(4)=−2.Which of the following is guaranteed by the Intermediate Value Theorem?Choose 1 answer:(A) h(c)=−1 for at least one c between −2 and 2(B) h(c)=3 for at least one c between −2 and 2(C) h(c)=−1 for at least one c between 0 and 4(D) h(c)=3 for at least one c between 0 and 4
Q. Let h be a continuous function on the closed interval [0,4], where h(0)=2 and h(4)=−2.Which of the following is guaranteed by the Intermediate Value Theorem?Choose 1 answer:(A) h(c)=−1 for at least one c between −2 and 2(B) h(c)=3 for at least one c between −2 and 2(C) h(c)=−1 for at least one c between 0 and 4(D) h(c)=3 for at least one c between 0 and 4
Theorem Statement: The Intermediate Value Theorem states that if a function f is continuous on a closed interval [a,b] and N is any number between f(a) and f(b), then there exists at least one c in the interval (a,b) such that f(c)=N.
Given Function Values: We are given that h(0)=2 and h(4)=−2. This means that the function h changes from a positive value to a negative value over the interval [0,4].
Application of Theorem: Since −1 is a value between 2 and −2, by the Intermediate Value Theorem, there must be at least one value c in the interval (0,4) such that h(c)=−1.
Verification of Another Value: The value 3 is not between 2 and −2, so the Intermediate Value Theorem does not guarantee that there is a value c in the interval (0,4) such that h(c)=3.
Correct Answer: Therefore, the correct answer is that h(c)=−1 for at least one c between 0 and 4, which corresponds to choice (C).
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