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 ff is continuous on a closed interval [a,b][a, b] and takes on values f(a)f(a) and f(b)f(b), then for any value yy between f(a)f(a) and f(b)f(b), there exists at least one c(a,b)c \in (a, b) such that f(c)=yf(c) = y.

In your case:

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

According to the IVT, for any value yy between 1010 and 2020, there must be some c(2,4)c \in (2, 4) such that f(c)=yf(c) = y.

What is guaranteed by the IVT:

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

  • f(c)=15f(c) = 15,
  • f(c)=12f(c) = 12,
  • f(c)=18f(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 ff were not continuous on the interval?
  5. What happens if f(a)f(a) and f(b)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.

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Math Problem Analysis

Mathematical Concepts

Continuity
Intermediate Value Theorem

Formulas

-

Theorems

Intermediate Value Theorem

Suitable Grade Level

Grades 10-12