From the image, the problem involves analyzing the graph of a function $f(x)$ and answering several questions regarding its increasing and decreasing intervals, local maxima and minima, and absolute extrema. Let's break down each part of the question:

(a) Increasing Intervals

To determine where $f(x)$ is increasing, we look for sections of the graph where the function is going upwards as $x$ increases.

From the graph:

• $f(x)$ increases from $x = -3$ to $x = -1$.
• It also increases from $x = 3$ to $x = 6$.

Thus, the intervals are: $(-3, -1), (3, 6)$

(b) Decreasing Intervals

For decreasing intervals, the graph goes downwards as $x$ increases.

From the graph:

• $f(x)$ decreases from $x = -1$ to $x = 3$.
• It also decreases from $x = 6$ to $x = 9$.

Thus, the intervals are: $(-1, 3), (6, 9)$

(c) Local Maxima

Local maxima occur at points where the graph changes from increasing to decreasing.

From the graph:

• There is a local maximum at $x = -1$ with $f(x) = 12$.

Thus, the local maximum is at $f(x) = 12$.

(d) Local Minima

Local minima occur at points where the graph changes from decreasing to increasing.

From the graph:

• There is a local minimum at $x = 3$ with $f(x) = -12$.

Thus, the local minimum is at $f(x) = -12$.

(e) Absolute Maximum

The absolute maximum is the highest point on the graph.

• The highest point is at $x = -1$ with $f(x) = 12$.

Thus, the absolute maximum is $f(x) = 12$.

(f) Absolute Minimum

The absolute minimum is the lowest point on the graph.

• The lowest point is at $x = 3$ with $f(x) = -12$.

Thus, the absolute minimum is $f(x) = -12$.

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

Here are 5 related questions to deepen your understanding:

1. How do you determine increasing and decreasing intervals from a graph?
2. What is the difference between a local maximum and an absolute maximum?
3. Can a function have more than one absolute minimum or maximum?
4. How can you verify the local extrema using the first derivative test?
5. Why are open intervals used to describe increasing and decreasing regions?

Tip: Always check for changes in direction (increasing or decreasing) to identify local maxima and minima on a graph.