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To determine the truth values for each statement, let's analyze each one based on the properties of \( f(x) \) inferred from the graph of \( f'(x) \):

1. **Statement (i): \( f(x) \) has a local maximum at \( x = -3 \)**  
   - A local maximum for \( f(x) \) occurs where \( f'(x) \) changes from positive to negative. In the graph, around \( x = -3 \), \( f'(x) \) changes from positive to negative. This suggests that \( f(x) \) has a local maximum at \( x = -3 \).
   - **Conclusion: True (1)**

2. **Statement (ii): \( f(x) \) has neither a local minimum nor maximum at \( x = 3 \)**  
   - At \( x = 3 \), \( f'(x) = 0 \) (since the graph of \( f'(x) \) touches the x-axis at \( x = 3 \)), but it does not change sign; \( f'(x) \) remains negative before and after \( x = 3 \). This indicates a horizontal inflection point, meaning there is neither a maximum nor a minimum for \( f(x) \) at \( x = 3 \).
   - **Conclusion: True (1)**

3. **Statement (iii): \( f(x) \) has a local minimum near \( x = -2 \)**  
   - A local minimum for \( f(x) \) would occur where \( f'(x) \) changes from negative to positive. However, near \( x = -2 \), the graph of \( f'(x) \) does not change from negative to positive; instead, it remains negative. This implies there is no local minimum near \( x = -2 \).
   - **Conclusion: False (2)**

Based on this analysis, the answers for the statements (in the specified order) are:

**1, 1, 2**

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

1. Why does the sign change in \( f'(x) \) indicate a local maximum or minimum in \( f(x) \)?
2. How can you determine the exact location of inflection points based on \( f'(x) \) alone?
3. What does it mean when \( f'(x) \) remains constant (doesn’t change sign) around a particular point?
4. How would the answer change if \( f(x) \) were discontinuous at any point in the domain?
5. What additional information could we derive if we had the graph of \( f''(x) \)?

**Tip:** Remember that if \( f'(x) \) is zero but does not change sign, it generally indicates an inflection point rather than a local extremum.

Suppose that f(x) is continuous, and f'(x) is given in the figure below. Determine the truth of the statements based on the graph of f'(x): (i) f(x) has a local maximum at x = -3, (ii) f(x) has neither a local minimum nor maximum at x = 3, (iii) f(x) has a local minimum near x = -2.

Analyze the graph of f'(x) to determine local maxima and minima for f(x). This problem uses the first derivative test to identify critical points and assess the truth of statements about local extremum points at x = -3, x = 3, and near x = -2. Suitable for college-level calculus students.

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