Let's analyze the graph to answer your questions:

1. On what intervals is $f(x)$ increasing? A function is increasing when its derivative is positive, i.e., when the graph is sloping upward. Based on the graph:

   • $f(x)$ appears to increase on the intervals $(-5, -3)$, $(1, 3)$, and $(5, 6)$.

2. On what intervals is $f(x)$ decreasing? A function is decreasing when its derivative is negative, i.e., when the graph is sloping downward. From the graph:

   • $f(x)$ decreases on $(-6, -5)$, $(-3, 1)$, and $(3, 5)$.

3. At what values of $x$ does $f(x)$ have a local maximum? A local maximum occurs where the graph changes from increasing to decreasing (i.e., a peak). In the graph:

   • There is a local maximum at approximately $x = -3$ and $x = 3$.

4. At what values of $x$ does $f(x)$ have a local minimum? A local minimum occurs where the graph changes from decreasing to increasing (i.e., a trough). From the graph:

   • There is a local minimum at approximately $x = -5$ and $x = 1$.

Would you like any further clarification or details?

Here are five questions that expand on this:

1. How do you find the derivative of a function to confirm increasing/decreasing intervals?
2. What conditions define a local maximum and a local minimum in terms of derivatives?
3. Can you estimate the exact values of local maxima and minima using calculus?
4. How could concavity or inflection points be determined from this graph?
5. What real-world problems can be modeled by functions with multiple maxima and minima?

Tip: For deeper analysis, calculate the first derivative of the function to confirm where it's zero or undefined—these are potential extrema!