To help solve the problem, let's analyze the graph of the function $f$ shown in the image and answer the questions step by step.

1. Intervals of Increasing and Decreasing:

   • The function is increasing where the slope of the curve is positive (where it is going uphill).
   • The function is decreasing where the slope is negative (where it is going downhill).

   From the graph:

   • The function appears to be increasing from $x \approx -4$ to $x = 0$ and then from $x \approx 2$ onwards.
   • The function is decreasing from $x = 0$ to $x \approx 2$.

   Therefore:

   • Increasing on $(-4, 0) \cup (2, \infty)$
   • Decreasing on $(0, 2)$

2. Local Maximum and Local Minimum:

   • A local maximum is where the graph changes from increasing to decreasing. This happens at $x = 0$.
   • A local minimum is where the graph changes from decreasing to increasing. This occurs at $x \approx 2$.

   Therefore:

   • Local maximum at $x = 0$, $f(0) = 20$ (estimated from the graph).
   • Local minimum at $x \approx 2$, $f(2) = -12$ (estimated from the graph).

Let me know if you need further clarification!

Related Questions:

1. How do we calculate the exact coordinates of critical points (maxima and minima)?
2. What is the derivative of a function, and how is it used to determine increasing/decreasing intervals?
3. How can we check if a critical point is a maximum or minimum using the second derivative?
4. What is the significance of the inflection point on the graph?
5. How do different types of functions behave in terms of increasing and decreasing intervals?

Tip: Remember that critical points occur where the derivative is zero or undefined. Use this to find turning points!