Math Problem Statement
Let F be a function that is continuous on the closed interval [2,4] with f(2) =10 and f(4) = 20. Which of the following is guaranteed by the intermediate value theorem?
Solution
The Intermediate Value Theorem (IVT) states that if a function is continuous on a closed interval and takes on values and , then for any value between and , there exists at least one such that .
In your case:
- is continuous on the interval .
- and .
According to the IVT, for any value between and , there must be some such that .
What is guaranteed by the IVT:
For any value , there exists at least one such that . For example, it guarantees there is some such that:
- ,
- ,
- ,
- And any other value between 10 and 20.
Would you like more details on how the IVT works, or do you have any specific questions?
Relative questions:
- How does the Intermediate Value Theorem apply to differentiable functions?
- Can the IVT be applied to functions that are not continuous?
- What is an example of a real-world situation where the IVT can be used?
- How would you apply the IVT if were not continuous on the interval?
- What happens if and are the same value—does the IVT still apply?
Tip:
When using the Intermediate Value Theorem, always ensure the function is continuous on the entire interval.
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Math Problem Analysis
Mathematical Concepts
Continuity
Intermediate Value Theorem
Formulas
-
Theorems
Intermediate Value Theorem
Suitable Grade Level
Grades 10-12
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