Let h be a continuous function on the closed interval [-3,4], where h(-3)=-1 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 -3 and 4 (B) h(c)=-2 for at least one c between -3 and 4 (C) h(c)=-2 for at least one c between -1 and 2 (D) h(c)=1 for at least one c between -1 and 2

Question

Let  h be a continuous function on the closed interval  [-3,4], where  h(-3)=-1 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 -3 and 4 (B)  h(c)=-2 for at least one  c between -3 and 4 (C)  h(c)=-2 for at least one  c between -1 and 2 (D)  h(c)=1 for at least one  c between -1 and 2

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Theorem Application: The Intermediate Value Theorem states that if a function 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. We need to apply this theorem to the function h and the given values.Endpoint Values: First, we check the value of h at the endpoints of the interval. We have h(-3) = -1 and h(4) = 2. This means that the function h takes on the values -1 and 2 at the ends of the interval [-3, 4].Option (A) Analysis: Now, we look at the options given and apply the Intermediate Value Theorem. Option (A) suggests that h(c) = 1 for some c between -3 and 4. Since 1 is between -1 and 2, the Intermediate Value Theorem guarantees that there is at least one c in the interval [-3, 4] such that h(c) = 1.Option (B) Analysis: Option (B) suggests that h(c) = -2 for some c between -3 and 4. However, since -2 is not between h(-3) = -1 and h(4) = 2, the Intermediate Value Theorem does not guarantee that there is a c in the interval [-3, 4] such that h(c) = -2.Option (C) Consideration: Option (C) is not applicable because the Intermediate Value Theorem requires the interval to be the same as where the function is continuous and the values are given. Since we are given the interval [-3, 4] and not [-1, 2], we cannot apply the theorem to the interval [-1, 2] directly.Option (D) Consideration: Option (D) is similar to option (C) in that it refers to an interval [-1, 2] which is not the interval we are considering for the application of the Intermediate Value Theorem. Therefore, it cannot be guaranteed by the theorem based on the information given.

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