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tell me all intervals where f is decreasing

To determine the intervals where the function \( f(x) \) is decreasing, we need to look at the graph and identify the sections where the slope of the function is negative. In other words, we look for the portions of the graph where the function is moving downward as we move from left to right.

### Analysis of the Graph:

1. **First Interval:**
   - The function \( f(x) \) starts increasing until it reaches a local maximum. After that, it begins to decrease.
   - This happens from approximately \( x = 0 \) to \( x = 1 \).

2. **Second Interval:**
   - After the first local minimum around \( x = 1 \), the function increases again until it reaches another local maximum around \( x = 3 \). Then it decreases again.
   - This second decreasing interval is from approximately \( x = 3 \) to \( x = 4 \).

### Intervals Where \( f(x) \) is Decreasing:
- **Interval 1:** \( 1 < x < 2 \)
- **Interval 2:** \( 3 < x < 4 \)

These intervals represent the sections where the function \( f(x) \) is decreasing.

Would you like more details or have any other questions? Here are five follow-up questions to consider:

1. How can you find the exact values of \( x \) where the function has local maxima and minima?
2. What is the significance of increasing and decreasing intervals in the context of optimization problems?
3. How does the derivative of the function relate to these intervals of increase and decrease?
4. Can you determine the concavity of the function based on the graph?
5. What does the overall shape of this graph suggest about the degree of the polynomial?

**Tip:** To find exact intervals of increase and decrease algebraically, consider using the first derivative of the function.

Explore how to identify intervals where a function decreases through graphical analysis and the first derivative test. Understand the concepts of derivatives and function behavior. Suitable for advanced high school and college students.

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