https://file.aiquickdraw.com/mathbot/file/17306883895295cxc8x3u.jpg

To solve this problem, let’s analyze each statement based on the graph of \( f'(x) \) provided:

1. **Statement (i): \( f(x) \) has a local maximum at \( x = 1 \)**  
   - To have a local maximum at \( x = 1 \), \( f(x) \) should change from increasing to decreasing at that point. This means \( f'(x) \) would go from positive to negative as \( x \) crosses 1.
   - From the graph of \( f'(x) \), we see that \( f'(x) \) crosses the x-axis from positive to negative at \( x = 1 \). This indicates that \( f(x) \) has a local maximum at \( x = 1 \).
   - **Statement (i) is True.**

2. **Statement (ii): \( f(x) \) must have a local minimum or maximum at \( x = 3 \)**  
   - A local minimum or maximum at \( x = 3 \) would require \( f'(x) \) to be zero or undefined there.
   - On the graph, \( f'(x) \) touches the x-axis at \( x = 3 \), meaning \( f'(3) = 0 \). However, \( f'(x) \) does not change sign around \( x = 3 \); it remains negative on both sides.
   - Since there is no sign change, \( x = 3 \) is not a point of local extremum for \( f(x) \).
   - **Statement (ii) is False.**

3. **Statement (iii): \( f(x) \) has a local minimum at \( x = -3 \)**  
   - To confirm a local minimum at \( x = -3 \), \( f'(x) \) should change from negative to positive as \( x \) crosses -3.
   - Observing the graph, \( f'(x) \) crosses the x-axis from negative to positive at \( x = -3 \), indicating a local minimum for \( f(x) \) at this point.
   - **Statement (iii) is True.**

### Conclusion:
The correct answers are:
- **(i) True**
- **(ii) False**
- **(iii) True**

So, the answer is: **1, 2, 1**

Would you like more details on any of these statements?

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### Follow-Up Questions:
1. What would it mean for \( f(x) \) if \( f'(x) \) is always positive on an interval?
2. How would you identify an inflection point by looking at the graph of \( f'(x) \)?
3. Why is it significant that \( f'(x) \) touches but does not cross the x-axis at \( x = 3 \)?
4. What additional information could we derive about \( f(x) \) from a graph of \( f''(x) \)?
5. How do the critical points of \( f(x) \) relate to the zeros of \( f'(x) \)?

### Tip:
When analyzing the graph of \( f'(x) \), remember that a zero crossing of \( f'(x) \) with a change in sign indicates a local extremum (maximum or minimum) in \( f(x) \).

Suppose that f(x) is continuous, and f'(x) is given in the figure below.
Consider the following statements:
(i) f(x) has a local maximum at x = 1
(ii) f(x) must have a local minimum or maximum at x = 3
(iii) f(x) has a local minimum at x = -3
Determine which of the above statements are True (1) or False (2).

Explore how to determine local maximum and minimum points using the first derivative test. This problem analyzes the function's derivative graph to determine if f(x) has a local maximum at x = 1, a local minimum or maximum at x = 3, and a local minimum at x = -3.

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