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What is the intervals of increasing and decreasing

To determine the intervals of increasing and decreasing from the graph you've uploaded, we need to analyze the behavior of the function's slope over different x-intervals.

### Steps:
1. **Identify critical points:**
   - Peaks (local maxima)
   - Valleys (local minima)
   
2. **Determine the intervals between the critical points:**
   - An increasing function has a positive slope (going up from left to right).
   - A decreasing function has a negative slope (going down from left to right).

### Analysis of the graph:

- The **local maximum** is at \((-5.5, 632.813)\).
- The **local minimum** is at \((-1.5, -392)\).
- The graph starts decreasing from \(x = -\infty\) and reaches a minimum at \((-5.5, 632.813)\), and then it decreases toward the local minimum \((-1.5, -392)\).
- There seems to be another turning point around \((2, 0)\), where it increases briefly again and then decreases.

### Based on the graph:

- **Intervals of increase** (where the graph rises):
  - From \(x = -\infty\) to \(x = -5.5\) (until it reaches the local max).
  - From \(x = -1.5\) to around \(x = 2\).
  
- **Intervals of decrease** (where the graph falls):
  - From \(x = -5.5\) to \(x = -1.5\) (until it reaches the local min).
  - From \(x = 2\) onwards (after the peak around \(x = 2\)).

Would you like more details on how to find these intervals algebraically, or have any questions about the graph?

Here are 5 questions you can explore next:
1. What are the exact x-coordinates of critical points using the derivative?
2. How can you algebraically verify the intervals of increase and decrease?
3. What is the second derivative's role in determining concavity?
4. Can we find the equation of the function from this graph?
5. What are the real-life applications of identifying increasing and decreasing intervals in a function?

**Tip:** Critical points occur where the derivative equals zero or is undefined.

What is the intervals of increasing and decreasing?

Learn how to determine the intervals of increasing and decreasing for a function based on its critical points. Analyze the function graph to identify local maxima and minima and apply the First Derivative Test. This problem is suitable for students in Grades 11-12.

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