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![Let
f be a continuous function on the closed interval
[-2,1], where
f(-2)=3 and
f(1)=6.
Which of the following is guaranteed by the Intermediate Value Theorem?
Choose 1 answer:
(A)
f(c)=0 for at least one
c between -2 and 1
(B)
f(c)=4 for at least one
c between -2 and 1
(c)
f(c)=0 for at least one
c between 3 and 6
(D)
f(c)=4 for at least one
c between 3 and 6](https://externalquestionsimg-a4fuc2b6e0gtdqge.z01.azurefd.net/external-question-images/images/images/ka/PreCalculus/Limits%20and%20continuity/Working%20with%20the%20intermediate%20value%20theorem/Q-0.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
- 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 . We need to determine which statement is guaranteed by the Intermediate Value Theorem.
- Option (A) Analysis: Option (A) suggests that for at least one between and . However, since and , and is not between and , the Intermediate Value Theorem does not guarantee that will be for any in the interval .
- Option (B) Analysis: Option (B) suggests that for at least one between and . Since is between and , the Intermediate Value Theorem guarantees that there will be at least one in the interval such that .
- Option (C) Analysis: Option (C) suggests that for at least one between and . This option is not relevant because the values and are the function values, not the values in the domain of .
- Option (D) Analysis: Option (D) suggests that for at least one between and . This option is also not relevant because, again, and are the function values, not the values in the domain of .
