Let g be a continuous function on the closed interval [−1,4], where g(−1)=−4 and g(4)=1.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 4(B) g(c)=3 for at least one c between −4 and 1(C) g(c)=−3 for at least one c between −4 and 1(D) g(c)=3 for at least one c between −1 and 4
Q. Let g be a continuous function on the closed interval [−1,4], where g(−1)=−4 and g(4)=1.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 4(B) g(c)=3 for at least one c between −4 and 1(C) g(c)=−3 for at least one c between −4 and 1(D) g(c)=3 for at least one c between −1 and 4
IVT Explanation: 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.
Given Function and Intervals: We are given that g is continuous on the closed interval [−1,4], g(−1)=−4, and g(4)=1. We need to find if there is a value c in the interval [−1,4] such that g(c) equals a certain value.
Option (A) Analysis: Option (A) suggests that g(c)=−3 for at least one c between −1 and 4. Since −3 is between g(−1)=−4 and g(4)=1, by the Intermediate Value Theorem, there must be at least one c in the interval [−1,4] where g(c)=−3.
Option (B) Analysis: Option (B) suggests that g(c)=3 for at least one c between −4 and 1. However, the interval [−4,1] is not the interval we are considering for the function g, which is defined on [−1,4]. Therefore, this option is not relevant to the given information.
Option (C) Analysis: Option (C) suggests that g(c)=−3 for at least one c between −4 and 1. Similar to option (B), this interval is not the one we are considering for the function g, which is defined on [−1,4]. Therefore, this option is also not relevant to the given information.
Option (D) Analysis: Option (D) suggests that g(c)=3 for at least one c between −1 and 4. Since 3 is not between g(−1)=−4 and g(4)=1, the Intermediate Value Theorem does not guarantee that there is a value c where g(c)=3 in the interval [−1,4].
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