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The graph shown in the image appears to represent a function resembling a rational function (likely of the form \( f(x) = \frac{1}{x^2} \) or something similar), with vertical asymptotes on both sides. We are tasked with identifying the intervals where the function is concave upward and concave downward.

### Analyzing the Concavity:

1. **Concave Downward**: 
   A function is concave downward when its second derivative is negative, which corresponds to the graph bending downward. This behavior is observed where the graph forms a "hill" or arches downward.

   - From the graph, it looks like the function is concave downward in the interval \( (-\infty, -2) \). The curve is arching downward in this region.
   - Similarly, it is concave downward in the interval \( (2, \infty) \), as the function also forms a downward arch.

2. **Concave Upward**:
   A function is concave upward when its second derivative is positive, corresponding to the graph bending upward, forming a "bowl."

   - The function is concave upward in the interval \( (-2, 2) \), as it forms a U-shape (bending upward).

### Conclusion:
- **Concave Upward**: \( (-2, 2) \)
- **Concave Downward**: \( (-\infty, -2) \cup (2, \infty) \)

Would you like further details on how concavity is determined through derivatives?

### Relative Questions:
1. What is the relationship between the first and second derivative and concavity?
2. How can we use the second derivative test to find points of inflection?
3. What are some common types of functions that exhibit both concave upward and downward behavior?
4. How does concavity change near vertical asymptotes?
5. Could the graph change concavity at points of discontinuity?

### Tip:
To find points of inflection (where concavity changes), you can solve \( f''(x) = 0 \) or determine where \( f''(x) \) changes sign.

Identify the open intervals on which the graph of the function is concave upward or concave downward. Assume that the graph extends past what is shown.

Analyze the graph of a rational function to determine the intervals where the function is concave upward and concave downward. Learn how to apply the second derivative test to identify concavity. Suitable for students in Grades 11-12.

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