Let's analyze the graph and the options to find the correct answer.

1. Intervals of Increase and Decrease:

   • The function decreases from the far left (negative infinity) until $x = 0$.
   • Then, it increases from $x = 0$ to $x = 1$, as the graph moves upward.
   • The function decreases again from $x = 1$ to $x = 2$, where it reaches a low point.
   • Finally, the function increases from $x = 2$ onwards towards positive infinity.

2. Local Minima and Maxima:

   • At $x = 0$, the function has a local minimum (the lowest point in that region).
   • At $x = 1$, the function has a local maximum (a peak).
   • At $x = 2$, the function has another local minimum.

Now, let's match these findings with the given options:

• Option a states:

  • Decreasing on $(-\infty, 0) \cup (1, 2)$ ✔️
  • Increasing on $(0, 1) \cup (2, \infty)$ ✔️
  • Local minima at $x = 0$ and $x = 2$ ✔️
  • Local maximum at $x = 1$ ✔️

• Option b has an incorrect local minimum of $-2$, so it is incorrect.

• Option c and d also incorrectly state the local minimum as $-2$, making them incorrect as well.

Thus, Option a is the correct answer.

Would you like further clarification on this? Here are five related questions for deeper understanding:

1. How do you determine intervals of increase and decrease from a graph?
2. What is the difference between a local minimum and a global minimum?
3. How would you algebraically find critical points of a function?
4. What is the significance of concavity in determining local extrema?
5. How does the derivative of a function relate to increasing and decreasing intervals?

Tip: Always check both increasing/decreasing intervals and the location of local maxima/minima to avoid common mistakes when analyzing graphs.