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The question is asking to apply the **Intermediate Value Theorem (IVT)** to the given function \( f(x) = 2x^3 + 5x^2 + 8 \) on the interval \([0,1]\). The IVT states that if a function is continuous on a closed interval, and if \( f(a) \neq f(b) \), then for any \( k \) between \( f(a) \) and \( f(b) \), there exists a number \( c \) in the interval \([a, b]\) such that \( f(c) = k \).

Let's apply the steps:

1. First, calculate \( f(0) \):
   \[
   f(0) = 2(0)^3 + 5(0)^2 + 8 = 8
   \]

2. Next, calculate \( f(1) \):
   \[
   f(1) = 2(1)^3 + 5(1)^2 + 8 = 2 + 5 + 8 = 15
   \]

Now, the values of \( f(x) \) at the endpoints of the interval \([0, 1]\) are:
- \( f(0) = 8 \)
- \( f(1) = 15 \)

By the IVT, any value \( k \) that lies between \( f(0) = 8 \) and \( f(1) = 15 \) will have a corresponding \( c \) in the interval \([0, 1]\) such that \( f(c) = k \).

Thus, the values of \( k \) that satisfy the Intermediate Value Theorem are:
\[
8 \leq k \leq 15
\]

This is the range for \( k \).

Would you like further details on how the Intermediate Value Theorem works or on any part of this solution?

Here are five related questions to deepen your understanding:
1. What happens if \( k \) is outside the interval \([8, 15]\)?
2. Can we apply the IVT to a discontinuous function?
3. What does the IVT guarantee about the number of solutions for \( f(c) = k \)?
4. How can the IVT be used to approximate solutions to equations?
5. How can you verify that a function is continuous on a given interval?

**Tip:** The Intermediate Value Theorem only guarantees the existence of a solution, not how many or exactly where it is located.

Consider the function f(x) = 2x^3 + 5x^2 + 8. For what values of the number k does the Intermediate Value Theorem tell us that there is a number c in the interval [0, 1] such that f(c) = k?

Learn how to apply the Intermediate Value Theorem to the function f(x) = 2x^3 + 5x^2 + 8 within the interval [0, 1]. This problem explores key concepts in calculus, showing how to find values of k where f(c) = k. Suitable for Grades 10-12.

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