https://file.aiquickdraw.com/mathbot/file/1729577201277iu6mc41c.jpg

What is the answer from the provided otpions

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.

Use the graph to determine the open intervals over which f(x) is increasing, decreasing, or constant, and then determine all the local minimum and maximum values on the graph.

Analyze the graph to determine intervals where f(x) is increasing or decreasing and identify local minima and maxima. Learn key concepts such as increasing and decreasing functions, local extrema, and graphical interpretations of these characteristics.

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