From the image you uploaded, you are being asked two questions about the function $f(x)$, which is represented graphically:

1. Identify the critical numbers of $f$.

   • Critical numbers occur where the derivative $f'(x) = 0$ or is undefined, corresponding to points where the slope of the tangent line is horizontal (peaks, troughs) or there is a sharp corner.
   • From the graph, it looks like the derivative is zero at the local maximum and minimum points. Based on observation, the peaks and troughs are around $x = -3$, $x = 0$, and $x = 3$. So, the critical numbers should be $x = -3, 0, 3$.

2. Identify the open intervals on which $f$ is increasing or decreasing.

   • A function is increasing where the slope is positive (upward trend) and decreasing where the slope is negative (downward trend).
   • From the graph:
     • $f(x)$ is increasing on the intervals $(-\infty, -3)$ and $(0, 3)$.
     • $f(x)$ is decreasing on the intervals $(-3, 0)$ and $(3, \infty)$.

Would you like further explanation on these intervals or points? Feel free to ask any follow-up questions.

Here are 5 related questions:

1. How do you mathematically determine the critical points from a given function?
2. What are the conditions that define a local maximum or minimum?
3. What is the relationship between the first derivative and the increasing or decreasing nature of a function?
4. How can second derivatives help in analyzing the concavity of a graph?
5. Why is it important to check endpoints or boundaries for critical points in real-world applications?

Tip: Always check the derivative's sign changes around critical points to ensure proper classification (increasing or decreasing intervals).