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In this problem, you are given three functions (A, B, and C) and asked to match them with the appropriate graphs of their derivatives, \( f'(x) \), from the choices I to VIII.

Here’s how you can approach the problem:

### Step-by-step approach:

1. **Look at the critical points of \( f(x) \):**
   - The critical points (where \( f'(x) = 0 \)) occur at the peaks, troughs, or flat points of the graph of \( f(x) \). These will correspond to the points where the graph of \( f'(x) \) crosses the x-axis (i.e., where \( f'(x) = 0 \)).

2. **Analyze the behavior around critical points:**
   - If \( f(x) \) is increasing, \( f'(x) > 0 \).
   - If \( f(x) \) is decreasing, \( f'(x) < 0 \).

3. **Match this with the behavior in the derivative graphs:**
   - For each function, determine where \( f(x) \) is increasing, decreasing, or has a critical point, and match this behavior with the corresponding graph.

### Analysis of Functions:

#### Function A:
- **Shape**: Function A starts decreasing, then increases after a minimum point.
- **Derivative behavior**:
  - Decreasing means \( f'(x) < 0 \) on the left.
  - After the minimum point, \( f'(x) > 0 \).
  - There is a critical point (where \( f'(x) = 0 \)) at the minimum.
  
  **Matching Derivative**: This behavior matches Graph V, which shows \( f'(x) \) going from negative to positive with a zero at a critical point.

#### Function B:
- **Shape**: Function B is a straight line, constantly decreasing.
- **Derivative behavior**:
  - Constantly decreasing means \( f'(x) \) is a constant negative value.

  **Matching Derivative**: This matches Graph VII, which shows a constant negative value for \( f'(x) \).

#### Function C:
- **Shape**: Function C is increasing, reaching a maximum, and then decreasing.
- **Derivative behavior**:
  - Increasing means \( f'(x) > 0 \).
  - Decreasing means \( f'(x) < 0 \).
  - There is a critical point (where \( f'(x) = 0 \)) at the maximum.
  
  **Matching Derivative**: This behavior matches Graph VI, which shows \( f'(x) \) going from positive to negative with a zero at the critical point.

### Conclusion:
- **Function A**: V
- **Function B**: VII
- **Function C**: VI

Would you like further clarifications or explanations on any of these steps?

Here are some additional related questions:

1. How do you determine where a function has a critical point based on its graph?
2. How does the second derivative affect the concavity of a function?
3. What is the significance of the inflection points in relation to the graph of a function and its derivative?
4. Can a function have a horizontal tangent without having a critical point?
5. How do we interpret the behavior of the derivative for linear functions?

**Tip**: Always remember that the derivative's graph gives you the slope of the original function at every point. If you master reading slope changes, you'll master derivatives!

Match the functions with their corresponding derivatives based on the graph choices I to VIII.

This problem involves matching functions with their corresponding derivatives by analyzing critical points and behavior from a given set of graph choices. Students will use their knowledge of derivatives and graph behavior to solve the problem. Suitable for advanced high school students.

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