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The problem asks you to determine the inflection points of \( f(x) \), given the graph of its second derivative \( f''(x) \).

### Steps to solve:

1. **Inflection Points**: Inflection points of \( f(x) \) occur where \( f''(x) = 0 \) and where the concavity of \( f(x) \) changes, meaning \( f''(x) \) changes sign.
   
2. **Identify Zeros of \( f''(x) \)**: From the graph, we can see that \( f''(x) = 0 \) at \( x = -7 \), \( x = 0 \), and \( x = 5 \). These are the potential points of inflection.

3. **Check Sign Changes**:
   - Near \( x = -7 \): \( f''(x) \) changes from positive to negative, indicating a concavity change.
   - Near \( x = 0 \): \( f''(x) \) changes from negative to positive.
   - Near \( x = 5 \): \( f''(x) \) changes from negative to positive.

Since all three points have sign changes, they are inflection points.

### Final Answers:
- **Number of inflection points**: 3
- **Inflection points**: \( x = -7, 0, 5 \)

Would you like further details or have any questions? Here are five related questions for practice:

1. What does the sign of \( f''(x) \) tell you about the concavity of \( f(x) \)?
2. If \( f''(x) \) does not change sign at a zero, is it an inflection point?
3. How can you verify inflection points using \( f'(x) \)?
4. What is the relationship between critical points of \( f(x) \) and inflection points?
5. How can the first derivative \( f'(x) \) help in identifying points of inflection?

**Tip**: Always check for sign changes in \( f''(x) \) when determining inflection points, as zeros alone do not guarantee concavity changes!

Based on the graph of the second derivative of f(x), determine the number of inflection points of f(x). You may assume f''(x) is continuous, f''(x) = 0 only at x = -7, x = 0, and x = 5.

Learn how to determine the number and location of inflection points based on the graph of the second derivative f''(x). The problem covers key calculus concepts such as concavity and inflection points using the second derivative test.

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