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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
- IVT Explanation: 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 . We are given that is continuous on the interval , , and .
- Option Analysis (A): We need to determine which of the given options is guaranteed by the Intermediate Value Theorem. Let's analyze each option:(A) for at least one between and : Since is not between and , the theorem does not guarantee a such that in the interval .
- Option Analysis (B): (B) for at least one between and : Since is between and , the theorem guarantees that there is at least one in the interval such that .
- Option Analysis (C): for at least one between and : This option is not relevant because the interval between and does not correspond to the -values but to the -values (function values). The theorem applies to the -values in the interval .
- Option Analysis (D): (D) for at least one between and : Similar to option (C), this option is not relevant because it refers to the -values. The theorem guarantees the existence of a for a value between and , not between and .
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