Let g be a continuous function on the closed interval [−1,3], where g(−1)=−2 and g(3)=−5.Which of the following is guaranteed by the Intermediate Value Theorem?Choose 1 answer:(A) g(c)=−3 for at least one c between −1 and 3(B) g(c)=−3 for at least one c between −5 and −2(C) g(c)=0 for at least one c between −5 and −2(D) g(c)=0 for at least one c between −1 and 3
Q. Let g be a continuous function on the closed interval [−1,3], where g(−1)=−2 and g(3)=−5.Which of the following is guaranteed by the Intermediate Value Theorem?Choose 1 answer:(A) g(c)=−3 for at least one c between −1 and 3(B) g(c)=−3 for at least one c between −5 and −2(C) g(c)=0 for at least one c between −5 and −2(D) g(c)=0 for at least one c between −1 and 3
The Intermediate Value Theorem: 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 number c in the interval (a,b) such that f(c)=N.
Given Function and Interval: We are given that g is continuous on the closed interval [−1,3], g(−1)=−2, and g(3)=−5. We need to find if there is a value c in the interval [−1,3] such that g(c) equals a certain value.
Option (A) Explanation: Option (A) suggests that g(c)=−3 for at least one c between −1 and 3. Since −3 is between g(−1)=−2 and g(3)=−5, by the Intermediate Value Theorem, there must be at least one c in the interval (−1,3) such that g(c)=−3.
Option (B) Explanation: Option (B) is incorrect because it refers to a range of c values between −5 and −2, which are y-values, not x-values. The Intermediate Value Theorem applies to x-values within the interval [−1,3].
Option (C) Explanation: Option (C) suggests that g(c)=0 for at least one c between −5 and −2, which is again referring to y-values, not x-values. This is not what the Intermediate Value Theorem guarantees.
Option (D) Explanation: Option (D) suggests that g(c)=0 for at least one c between −1 and 3. However, since 0 is not between g(−1)=−2 and g(3)=−5, the Intermediate Value Theorem does not guarantee that there is a c such that g(c)=0 in the interval [−1,3].
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