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(a). The figure depicts the graph of f(x) and the graph of its second derivative f''(x). Which of A and B is the graph of f(x)? Explain your answer.

To determine which of the graphs, A or B, represents \( f(x) \) and which represents its second derivative \( f''(x) \), let's analyze the characteristics of each graph.

### Understanding the Relationship between \( f(x) \) and \( f''(x) \):
1. **Concavity and \( f''(x) \)**:
   - When \( f(x) \) is concave up, \( f''(x) > 0 \).
   - When \( f(x) \) is concave down, \( f''(x) < 0 \).
   - When \( f(x) \) changes concavity (inflection point), \( f''(x) = 0 \).

2. **Behavior of \( f''(x) \) at Critical Points of \( f(x) \)**:
   - At a maximum or minimum of \( f(x) \), \( f''(x) \) might be positive (minimum) or negative (maximum), depending on whether the point is a local minimum or maximum.

### Analyzing the Graphs:
- **Graph A** has a region where it is concave down (around \( x = -0.5 \)) and a region where it is concave up (around \( x = 0.5 \)). The curve appears to change concavity, suggesting that it could have an inflection point. This is characteristic of \( f(x) \).

- **Graph B** consistently increases and changes sign, typical of a second derivative \( f''(x) \). The regions where it crosses the x-axis likely correspond to inflection points of \( f(x) \).

Given this information:

- **Graph A** represents \( f(x) \), the original function.
- **Graph B** represents \( f''(x) \), the second derivative of the function.

### Explanation:
- Graph A changes concavity, which corresponds to inflection points in the function \( f(x) \). Graph B crosses the x-axis at those points, indicating that the second derivative \( f''(x) \) changes sign, confirming that it is the graph of \( f''(x) \).

Would you like further details or have any questions about this explanation?

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Here are 5 related questions to consider:
1. How can the first derivative \( f'(x) \) be identified from the graphs of \( f(x) \) and \( f''(x) \)?
2. What is the significance of inflection points in the context of graph \( f(x) \)?
3. How would the graph change if \( f''(x) \) had no real roots?
4. What role does the first derivative play in determining the shape of the graph \( f(x) \)?
5. How can concavity help in sketching a rough graph of the original function if only \( f''(x) \) is known?

**Tip:** When analyzing graphs, always consider the relationship between the function, its first derivative, and its second derivative. This helps in understanding the overall behavior of the function and its changes.

Explore the relationship between the graph of a function f(x) and its second derivative f''(x) through an analysis of concavity and inflection points. Learn how to distinguish between the two graphs and understand their significance in calculus. Suitable for advanced high school students.

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