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![Let
f be a continuous function on the closed interval
[1,5], where
f(1)=1 and
f(5)=-3.
Which of the following is guaranteed by the Intermediate Value Theorem?
Choose 1 answer:
(A)
f(c)=-2 for at least one
c between 1 and 5
(B)
f(c)=-2 for at least one
c between -3 and 1
(C)
f(c)=2 for at least one
c between 1 and 5
(D)
f(c)=2 for at least one
c between -3 and 1](https://externalquestionsimg-a4fuc2b6e0gtdqge.z01.azurefd.net/external-question-images/images/images/ka/PreCalculus/Limits%20and%20continuity/Working%20with%20the%20intermediate%20value%20theorem/Q-8.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
- Apply Theorem to Function: The Intermediate Value Theorem states that if is a continuous function on a closed interval and is any number between and , then there exists at least one in the interval such that . We need to apply this theorem to the given function .
- Evaluate Function Endpoints: Since and , we know that the function values at the endpoints of the interval are and , respectively. The Intermediate Value Theorem will apply to any value that lies between and .
- Determine Guaranteed Value: Looking at the answer choices, we need to determine which value of is guaranteed to have a corresponding in the interval such that .
- Analyze Choice (A): Choice (A) suggests that for some between and . Since is between and , the Intermediate Value Theorem guarantees that there is at least one in the interval for which .
- Reject Choice (B): Choice (B) is incorrect because it refers to a between and , which is outside the interval we are considering.
- Reject Choice (C): Choice (C) suggests that for some between and . However, since is not between and , the Intermediate Value Theorem does not guarantee a in the interval such that .
- Reject Choice (D): Choice (D) is incorrect for the same reason as choice (B); it refers to a between and , which is outside the interval .
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