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?

The **Intermediate Value Theorem** (IVT) states that if a function \( f \) is continuous on a closed interval \([a, b]\) and takes on values \( f(a) \) and \( f(b) \), then for any value \( y \) between \( f(a) \) and \( f(b) \), there exists at least one \( c \in (a, b) \) such that \( f(c) = y \).

In your case:

- \( f \) is continuous on the interval \([2, 4]\).
- \( f(2) = 10 \) and \( f(4) = 20 \).

According to the IVT, for any value \( y \) between \( 10 \) and \( 20 \), there must be some \( c \in (2, 4) \) such that \( f(c) = y \).

### What is guaranteed by the IVT:
For any value \( y \in [10, 20] \), there exists at least one \( c \in (2, 4) \) such that \( f(c) = y \). For example, it guarantees there is some \( c \in (2, 4) \) such that:

- \( f(c) = 15 \),
- \( f(c) = 12 \),
- \( f(c) = 18 \),
- 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:
1. How does the Intermediate Value Theorem apply to differentiable functions?
2. Can the IVT be applied to functions that are not continuous?
3. What is an example of a real-world situation where the IVT can be used?
4. How would you apply the IVT if \( f \) were not continuous on the interval?
5. What happens if \( f(a) \) and \( f(b) \) 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.

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?

Explore how the Intermediate Value Theorem guarantees that a continuous function on [2,4] with f(2) = 10 and f(4) = 20 takes every value between 10 and 20. This detailed explanation is ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation