https://file.aiquickdraw.com/mathbot/file/1728482792685se2lp5qj.jpg

Let's go through each part of the problem step-by-step using the graph and description provided in the image.

### (i) Identify the critical numbers of \( f \)
Critical numbers occur where \( f' = 0 \) (horizontal tangent) or where \( f' \) is undefined.

Based on the graph:
1. \( f' \) changes direction (indicating a horizontal tangent) at \( x = -1 \), \( x = 0 \), and \( x = 2 \).
2. It also touches the x-axis at these points, indicating \( f' = 0 \).

So, the critical numbers of \( f \) are:
\[ x = -1, 0, 2 \]

### (ii) Identify the open interval(s) on which \( f \) is increasing or decreasing
To determine the intervals of increase or decrease, observe the behavior of \( f' \):
- When \( f' > 0 \), \( f \) is increasing.
- When \( f' < 0 \), \( f \) is decreasing.

By examining the graph:
- **Increasing**: \( f' > 0 \) on \( (-1, 0) \cup (2, 3) \)
- **Decreasing**: \( f' < 0 \) on \( (-\infty, -1) \cup (0, 2) \cup (3, \infty) \)

Therefore:
- Increasing: \( (-1, 0) \cup (2, 3) \)
- Decreasing: \( (-\infty, -1) \cup (0, 2) \cup (3, \infty) \)

### (iii) Determine whether \( f \) has a relative maximum, a relative minimum, or neither at each critical number
To determine if each critical point is a relative maximum, minimum, or neither, analyze the sign changes of \( f' \):

1. At \( x = -1 \):
   - \( f' \) changes from negative to positive, indicating a relative minimum.
   
2. At \( x = 0 \):
   - \( f' \) changes from positive to negative, indicating a relative maximum.
   
3. At \( x = 2 \):
   - \( f' \) changes from negative to positive, indicating a relative minimum.

So:
- Relative Minimum: \( x = -1, 2 \)
- Relative Maximum: \( x = 0 \)

### Summary of Answers:
1. **Critical numbers**: \( x = -1, 0, 2 \)
2. **Intervals of Increase/Decrease**:
   - Increasing: \( (-1, 0) \cup (2, 3) \)
   - Decreasing: \( (-\infty, -1) \cup (0, 2) \cup (3, \infty) \)
3. **Relative Extrema**:
   - Relative Minimum: \( x = -1, 2 \)
   - Relative Maximum: \( x = 0 \)

Would you like further clarification on any of these steps? Here are some related questions to consider:

1. What is the importance of identifying critical points in calculus?
2. How do you determine the concavity of a function using its second derivative?
3. What does the graph of the first derivative tell us about the function's behavior?
4. Can there be relative maxima or minima at points where the derivative does not exist?
5. How do we interpret inflection points based on the second derivative?

**Tip:** When analyzing derivatives, remember that intervals where the derivative is positive indicate increasing behavior of the original function, while intervals where it is negative indicate decreasing behavior.

Identify the critical numbers, open intervals of increase and decrease, and relative extrema for a function based on its derivative graph.

Learn how to find critical points, determine intervals of increase/decrease, and identify relative extrema using the graph of the first derivative. This problem focuses on key calculus concepts such as the first derivative test and critical points.

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