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![Let
g be a continuous function on the closed interval
[-3,3], where
g(-3)=0 and
g(3)=6.
Which of the following is guaranteed by the Intermediate Value Theorem?
Choose 1 answer:
(A)
g(c)=-2 for at least one
c between -3 and 3
(B)
g(c)=-2 for at least one
c between 0 and 6
(c)
g(c)=5 for at least one
c between 0 and 6
(D)
g(c)=5 for at least one
c between -3 and 3](https://externalquestionsimg-a4fuc2b6e0gtdqge.z01.azurefd.net/external-question-images/images/images/ka/PreCalculus/Limits%20and%20continuity/Working%20with%20the%20intermediate%20value%20theorem/Q-6.png)
Let be a continuous function on the closed interval , where and .Which of the following is guaranteed by the Intermediate Value Theorem?Choose answer:(A) for at least one between and (B) for at least one between and (C) for at least one between and (D) for at least one between and
Full solution
Q. Let be a continuous function on the closed interval , where and .Which of the following is guaranteed by the Intermediate Value Theorem?Choose answer:(A) for at least one between and (B) for at least one between and (C) for at least one between and (D) for at least one between and
- The Intermediate Value Theorem: The Intermediate Value Theorem states that if a function is continuous on a closed interval and is any number between and , then there exists at least one number in the interval such that .
- Given Function and Interval: We are given that is continuous on the closed interval , , and . This means that takes on every value between and at least once on the interval .
- Invalid Option (A): Option (A) suggests that for some between and . However, since and , the function does not take on values less than or greater than on the interval . Therefore, cannot be for any in the interval .
- Invalid Option (B): Option (B) is not valid because it refers to a range of values for (between and ) rather than a range of values for . The Intermediate Value Theorem applies to the values of in the domain, not the range of the function.
- Invalid Option (C): Option (C) is also not valid for the same reason as option (B). It refers to a range of values for (between and ) rather than a range of values for .
- Valid Option (D): Option (D) suggests that for some between and . Since and , and is a value between and , the Intermediate Value Theorem guarantees that there is at least one in the interval such that .
