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

To analyze the graph and determine the open intervals where the function is increasing, decreasing, or constant, as well as to identify the local minimum and maximum values, we'll follow these steps:

### 1. **Identify the behavior of the function:**
   - **Increasing**: The function is increasing on an interval where the graph is moving upwards as you go from left to right.
   - **Decreasing**: The function is decreasing on an interval where the graph is moving downwards as you go from left to right.
   - **Constant**: The function is constant on an interval where the graph is a horizontal line.

### 2. **Determine the open intervals:**
   - **Interval 1 (Decreasing)**: Starting from the leftmost part of the graph (likely from \(x \to -\infty\)) to the first local minimum, the function is decreasing.
   - **Interval 2 (Increasing)**: From the first local minimum to the local maximum, the function is increasing.
   - **Interval 3 (Decreasing)**: From the local maximum to the second local minimum, the function is decreasing.
   - **Interval 4 (Increasing)**: From the second local minimum onwards, the function is increasing.

### 3. **Identify local minima and maxima:**
   - **Local Maximum**: The highest point in the section of the graph where the function changes from increasing to decreasing.
   - **Local Minimum**: The lowest point in the section of the graph where the function changes from decreasing to increasing.

### **Detailed Analysis of the Graph:**
From the graph, let's identify the intervals more precisely:

- **Decreasing Interval 1**: From \(x \approx -\infty\) to \(x \approx -2\).
- **Increasing Interval 2**: From \(x \approx -2\) to \(x \approx 0\).
- **Decreasing Interval 3**: From \(x \approx 0\) to \(x \approx 2\).
- **Increasing Interval 4**: From \(x \approx 2\) to \(x \approx \infty\).

### **Local Extrema Values:**
- **Local Minimum 1**: Occurs at \(x \approx -2\), the value of the function at this point is the first local minimum.
- **Local Maximum**: Occurs at \(x \approx 0\), the value of the function at this point is the local maximum.
- **Local Minimum 2**: Occurs at \(x \approx 2\), the value of the function at this point is the second local minimum.

### **Summary of Results:**
- **Intervals of Increase**: \((-2, 0)\) and \((2, \infty)\).
- **Intervals of Decrease**: \((-\infty, -2)\) and \((0, 2)\).
- **Local Minimum Values**: At \(x \approx -2\) and \(x \approx 2\).
- **Local Maximum Value**: At \(x \approx 0\).

Would you like further details on any of these points, or do you have any other questions?

### 5 Relative Questions:
1. How can you determine the concavity of the function based on its graph?
2. What does the first derivative of this function represent in terms of the graph's behavior?
3. How could we use this graph to approximate the function's equation?
4. What role do the roots of the function play in determining the intervals of increase and decrease?
5. How can you confirm the exact coordinates of local minima and maxima algebraically?

### Tip:
When analyzing a function graphically, always check for smooth transitions between increasing and decreasing intervals to accurately identify local maxima and minima.

Explore how to analyze graphs to determine intervals of increase, decrease, and local extrema. This problem involves using the First Derivative Test and understanding graph behavior in calculus. Suitable for Grades 11-12 students.

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