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State the intervals where the following function is positive and negative

The graph you've provided seems to represent a cubic or higher-degree polynomial function with turning points. To determine where the function is positive or negative, we observe its behavior relative to the x-axis.

### Steps to analyze:
1. **Positive intervals**: The function is positive where the graph lies above the x-axis.
2. **Negative intervals**: The function is negative where the graph lies below the x-axis.

### Analysis from the graph:
- The function crosses the x-axis at approximately \( x = 0 \), \( x = 3 \), and \( x = 5 \).
- The function is positive between \( 0 < x < 3 \) and for \( x > 5 \), as it is above the x-axis in those regions.
- The function is negative between \( 3 < x < 5 \), where the curve lies below the x-axis.

### Intervals where the function is:
- **Positive**: \( (0, 3) \cup (5, \infty) \)
- **Negative**: \( (3, 5) \)

Would you like a more detailed explanation or have any further questions?

Here are 5 related questions:
1. What are turning points, and how do they affect the shape of polynomial graphs?
2. How can you determine the degree of a polynomial from its graph?
3. What does it mean when a function crosses the x-axis at multiple points?
4. How can you use calculus to find the exact points where a function changes from positive to negative?
5. What are the differences between intervals of increase and intervals where the function is positive?

**Tip**: To verify intervals analytically, finding the roots (where the function equals zero) helps break down the function's behavior.

State the intervals where the following function is positive and negative.

Analyze a polynomial function based on its graph and determine the intervals where the function is positive and negative. This example involves interpreting the function's behavior in relation to the x-axis and identifying key turning points. Suitable for students in Grades 10-12.

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