Let's go through the problem shown in the image.

Part (i): Identify the critical numbers of $f$

• Critical numbers occur where the derivative of the function is zero or undefined. These points typically correspond to peaks (local maxima), valleys (local minima), or points where the function's slope is flat.
• From the graph, the critical points appear to be at $x = -3$ and $x = 5$, where the function has horizontal tangent lines (flat spots at the peaks/valleys).

So the critical numbers are: $x = -3, 5$

Part (ii): Identify the intervals where $f$ is increasing or decreasing

• The function is increasing when the graph moves upward as you go from left to right. From the graph, this happens in the interval $(-3, 5)$.
• The function is decreasing when the graph moves downward as you go from left to right. From the graph, this happens in two intervals: $(-\infty, -3)$ and $(5, \infty)$.

So:

• Increasing: $(-3, 5)$
• Decreasing: $(-\infty, -3) \cup (5, \infty)$

Answer Summary:

• Critical numbers: $x = -3, 5$
• Increasing: $(-3, 5)$
• Decreasing: $(-\infty, -3) \cup (5, \infty)$

Would you like further details or clarifications?

Here are five follow-up questions you can explore:

1. How are critical numbers determined algebraically if we are given a function, not a graph?
2. What is the first derivative test, and how does it help in classifying critical points?
3. How can you find concave intervals on the graph?
4. What is the significance of the second derivative when analyzing graphs?
5. How do you determine if a critical point is a local maximum or minimum?

Tip: Always verify critical points by checking the first derivative and analyzing changes in sign to confirm whether the function is increasing or decreasing.