AI tutor
Welcome to Bytelearn!
Let’s check out your problem:
![Let
h be a continuous function on the closed interval
[-3,4], where
h(-3)=-1 and
h(4)=2.
Which of the following is guaranteed by the Intermediate Value Theorem?
Choose 1 answer:
(A)
h(c)=1 for at least one
c between -1 and 2
(B)
h(c)=1 for at least one
c between -3 and 4
(c)
h(c)=-2 for at least one
c between -3 and 4
(D)
h(c)=-2 for at least one
c between -1 and 2](https://externalquestionsimg-a4fuc2b6e0gtdqge.z01.azurefd.net/external-question-images/images/images/ka/PreCalculus/Limits%20and%20continuity/Working%20with%20the%20intermediate%20value%20theorem/Q-12.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
- Understand IVT: Understand the Intermediate Value Theorem (IVT). 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 in the interval such that .
- Apply to function : Apply the IVT to the given function . We are given that is continuous on the closed interval , , and . We need to find if there is a value in the interval such that equals a certain value.
- Evaluate based on IVT: Evaluate the choices based on the IVT.(A) for at least one between and . This choice is not relevant because the interval is not the interval we are considering for the function .
- Continue evaluating choices: Continue evaluating the choices.(B) for at least one between and . Since is between and , and the function is continuous on , the IVT guarantees that there is at least one in the interval where .
- Evaluate remaining choices: Evaluate the remaining choices.(C) for at least one between and . Since is not between and , the IVT does not guarantee that there is a where in the interval .
- Evaluate last choice: Evaluate the last choice.(D) for at least one between and . This choice is also not relevant because the interval is not the interval we are considering for the function , and is not between and .
