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?

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.