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The problem asks you to consider the function \( f(x) = 5x^3 + 2x^2 + 13 \) and determine for what values of \( k \) the Intermediate Value Theorem (IVT) guarantees that there exists a number \( c \) in the interval \([0, 1]\) such that \( f(c) = k \).

### Step 1: Evaluate the function at the endpoints \( x = 0 \) and \( x = 1 \)

- \( f(0) = 5(0)^3 + 2(0)^2 + 13 = 13 \)
- \( f(1) = 5(1)^3 + 2(1)^2 + 13 = 5 + 2 + 13 = 20 \)

So, the function takes values \( f(0) = 13 \) and \( f(1) = 20 \).

### Step 2: Apply the Intermediate Value Theorem (IVT)

The IVT states that if a function is continuous on a closed interval \([a, b]\) and takes values \( f(a) \) and \( f(b) \) at the endpoints, then for every value \( k \) between \( f(a) \) and \( f(b) \), there exists a \( c \in (a, b) \) such that \( f(c) = k \).

Since the function is a polynomial (and therefore continuous) on \([0, 1]\), we can conclude that for any \( k \) such that \( 13 \leq k \leq 20 \), there exists some \( c \in [0, 1] \) such that \( f(c) = k \).

### Final Answer:
The range of \( k \) is \( 13 \leq k \leq 20 \).

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Would you like more details, or do you have any questions about this solution?

Here are 5 related questions to explore:
1. How does the Intermediate Value Theorem apply to discontinuous functions?
2. Can you derive a general expression for the derivative of \( f(x) = 5x^3 + 2x^2 + 13 \)?
3. What is the significance of the continuity assumption in the IVT?
4. How would the solution change if the function were defined on a different interval, say \([0, 2]\)?
5. How can we numerically verify the existence of a \( c \) such that \( f(c) = k \) for some specific \( k \)?

**Tip**: When using the Intermediate Value Theorem, always check both the continuity of the function and the values at the endpoints of the interval.

Consider the function f(x) = 5x^3 + 2x^2 + 13. 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) = 5x^3 + 2x^2 + 13. This calculus problem covers polynomial functions and demonstrates how to find the range of k values for which a solution exists in the interval [0, 1].

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