Let's analyze the graph to answer the given questions:

a) List all the open intervals where $f$ is increasing:

In the graph, a function $f$ is said to be increasing where its slope is positive (i.e., the curve is going upwards from left to right). From observation:

1. The function $f$ increases in the interval from $(-\infty, -2)$ as the curve rises towards the local maximum at $(-2, 8)$.
2. It also increases between $(1, 3)$ as the curve rises towards the local maximum at $(3, 2)$.

So, the open intervals where $f$ is increasing are: $(-\infty, -2) \cup (1, 3)$

b) Identify local min/max of $f$:

From the graph, the local maxima and minima occur at the turning points where the slope of the function changes from positive to negative or vice versa.

• Local maximum:

  • At $(-2, 8)$, the function reaches a local maximum since the graph goes up to that point and then decreases.
  • At $(3, 2)$, the function also reaches a local maximum as the curve rises towards this point and then decreases.

• Local minimum:

  • At $(1, -3)$, the function reaches a local minimum as the curve falls to this point and then increases.

Thus, the local maxima are at $(-2, 8)$ and $(3, 2)$, and the local minimum is at $(1, -3)$.

Would you like more details or explanations on any part of this?

Here are five related questions to expand on the topic:

1. How can you determine where a function is increasing or decreasing from its derivative?
2. What is the significance of critical points in finding local extrema?
3. How would you use the second derivative test to classify local minima and maxima?
4. Can a function have more than two local maxima or minima?
5. How do concavity and inflection points relate to local extrema?

Tip: To determine where a function increases or decreases, you can find the derivative and solve for intervals where the derivative is positive (increasing) or negative (decreasing).