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Determine where the second derivative of the function is positive and where it is negative.

To determine where the second derivative of the function is positive or negative based on the given graph, we need to analyze the concavity of the curve. The second derivative, \( f''(x) \), gives information about the concavity:

- \( f''(x) > 0 \) (positive) indicates that the graph is concave up.
- \( f''(x) < 0 \) (negative) indicates that the graph is concave down.

From the image:
- **Concave up (second derivative positive):** This happens where the graph bends upwards, resembling a U-shape.
- **Concave down (second derivative negative):** This happens where the graph bends downwards, resembling an upside-down U-shape.

### Visual inspection of the graph:
- From \( x = -5 \) to approximately \( x = -2 \), the graph is bending downwards, meaning the second derivative is negative.
- From approximately \( x = -2 \) to \( x = 2 \), the graph changes to a concave-up shape, indicating that the second derivative is positive.
- Beyond \( x = 2 \), the graph shows signs of turning concave down again, suggesting the second derivative becomes negative.

### Summary:
- **\( f''(x) < 0 \) (negative):** From \( x = -5 \) to \( x = -2 \), and from \( x = 2 \) onwards.
- **\( f''(x) > 0 \) (positive):** From \( x = -2 \) to \( x = 2 \).

Would you like more details on how to calculate this analytically or any other questions on the graph? 

Here are 5 related questions:
1. How do you find the points of inflection on a graph?
2. What does the first derivative tell you about the graph?
3. How can you find critical points using the first derivative?
4. What is the relationship between concavity and the second derivative?
5. How do you interpret the behavior of the graph near turning points?

**Tip:** To analyze concavity changes more precisely, look for where the slope of the first derivative changes, as this corresponds to where the second derivative changes sign.

Explore how to determine where the second derivative of a function is positive or negative by analyzing the graph's concavity. This problem involves key calculus concepts, including the second derivative test and concavity behavior. Ideal for high school and college calculus students.

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