https://file.aiquickdraw.com/mathbot/file/1728482449695tjlab46d.jpg

Let's analyze the given graph and problem based on the provided image.

### (i) Identify the critical numbers of \( f \).
Critical numbers occur where the derivative \( f'(x) \) is either zero or undefined. Based on the graph, we see that the function changes direction at \( x = 1 \). This is a turning point where the slope of the tangent (or \( f'(x) \)) is zero. Therefore, the critical number is:

\[ x = 1 \]

### (ii) Identify the open interval(s) on which \( f \) is increasing or decreasing.

- **Decreasing interval**: The function decreases from \( x = -\infty \) to \( x = 1 \). Based on the graph, as the curve slopes downwards, the function is decreasing until the point where it reaches the critical number \( x = 1 \).
  
  Decreasing interval: \( (-\infty, 1) \)

- **Increasing interval**: The function starts increasing after \( x = 1 \). The curve slopes upward after passing through \( x = 1 \), meaning that the function is increasing from this point onward.
  
  Increasing interval: \( (1, \infty) \)

### (iii) Determine whether \( f \) has a relative maximum or minimum.

- **Relative minimum**: The graph shows a clear turning point at \( x = 1 \), where the function switches from decreasing to increasing, indicating a relative minimum.
  
  Relative minimum at \( x = 1 \).

- **Relative maximum**: The function does not have a relative maximum since the curve does not turn downward after increasing.

Relative maximum: None.

### Final Answers:
- **(i) Critical number**: \( x = 1 \)
- **(ii) Intervals**:
  - Increasing: \( (1, \infty) \)
  - Decreasing: \( (-\infty, 1) \)
- **(iii) Relative extrema**:
  - Relative minimum at \( x = 1 \)
  - No relative maximum.

Would you like more detailed explanations or have any questions about this solution?

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Here are 5 questions to expand your understanding:

1. What is the difference between a critical number and a point of inflection?
2. How can you confirm that \( x = 1 \) is a minimum using the second derivative test?
3. Can a function have more than one critical number?
4. How would the graph change if the critical number \( x = 1 \) were a maximum instead of a minimum?
5. What is the significance of a function's increasing and decreasing intervals?

**Tip**: Critical points are useful for identifying key behaviors of a function like maxima, minima, or saddle points. Always check the derivative around these points to understand the function's behavior.

Given a graph of a function f, identify the critical numbers, open intervals where f is increasing or decreasing, and determine whether f has a relative maximum or minimum.

Learn how to find critical numbers, increasing/decreasing intervals, and relative extrema of a function from its graph. This calculus problem involves applying the first derivative test to analyze the function's behavior.

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