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Let's break down the problem.

We are given the function \( f(x) = x^2 - 4x \) and asked to use the Intermediate Value Theorem (IVT) to determine if there is a solution to \( f(x) = 21 \) in the interval \([5, 8]\). If there is a solution, we are to find the value of \( c \) in the interval such that \( f(c) = 21 \).

### Steps:

1. **Check Continuity**:
   The function \( f(x) = x^2 - 4x \) is a polynomial, which means it is continuous on any interval, including \([5, 8]\). So, IVT can be applied.

2. **Evaluate the function at the endpoints of the interval**:
   - \( f(5) = 5^2 - 4(5) = 25 - 20 = 5 \)
   - \( f(8) = 8^2 - 4(8) = 64 - 32 = 32 \)

3. **Check if \( f(x) = 21 \) falls between \( f(5) = 5 \) and \( f(8) = 32 \)**:
   Since \( 21 \) is between 5 and 32, by the Intermediate Value Theorem, there exists at least one value \( c \in (5, 8) \) such that \( f(c) = 21 \).

4. **Find \( c \) such that \( f(c) = 21 \)**:
   Set \( f(c) = c^2 - 4c = 21 \). This simplifies to the quadratic equation:
   \[
   c^2 - 4c - 21 = 0
   \]
   Solve this using the quadratic formula:
   \[
   c = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-21)}}{2(1)} = \frac{4 \pm \sqrt{16 + 84}}{2} = \frac{4 \pm \sqrt{100}}{2} = \frac{4 \pm 10}{2}
   \]
   Thus, \( c = \frac{4 + 10}{2} = 7 \) or \( c = \frac{4 - 10}{2} = -3 \).

   Since we are looking for \( c \) in the interval \([5, 8]\), the only solution is \( c = 7 \).

### Answer:

The correct choice is **b.)** because:
- \( f(x) \) is continuous on \([5, 8]\),
- 21 is between \( f(5) = 5 \) and \( f(8) = 32 \),
- The correct \( c \) is 7.

Would you like more details or have any questions? 

Here are five related questions:
1. How does the Intermediate Value Theorem apply to real-world scenarios?
2. What other methods can be used to solve quadratic equations?
3. Can you explain the importance of continuity in the IVT?
4. How would this problem change if the function were not continuous?
5. What is the geometric interpretation of the IVT for this function?

**Tip**: Always check the continuity of a function before applying the IVT to ensure the theorem's conditions are satisfied!

Use the IVT to determine if there is a solution to f(x) = 21 in the interval between 5 and 8 for f(x) = x^2 - 4x. If so, find the value of c in the interval such that f(c) = 21.

Learn how to use the Intermediate Value Theorem to find a solution to f(x) = 21 in the interval [5, 8] for the quadratic function f(x) = x^2 - 4x. Includes a step-by-step guide and key concepts from calculus.

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