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To solve these exercises (33-36), we need to analyze the graphs of each function \( f(x) \) in terms of three aspects:

1. **Where \( f(x) \) is increasing or decreasing**: Look for intervals where the graph rises (increasing) or falls (decreasing) as you move from left to right.
2. **Where \( f(x) > 0 \)**: Identify intervals where the graph is above the x-axis.
3. **Where \( f(x) < 0 \)**: Identify intervals where the graph is below the x-axis.

I'll go through each problem based on these requirements.

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### 33
- **(a) Increasing**: The function is increasing when \( x \) is in \( (0, 2) \) and \( (4, 6) \).
- **(b) \( f(x) > 0 \)**: The function is positive (above the x-axis) for \( x \in (1, 5) \).
- **(c) \( f(x) < 0 \)**: The function is negative (below the x-axis) for \( x \in (-\infty, 1) \cup (5, \infty) \).

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### 34
- **(a) Increasing**: The function is increasing when \( x \) is in \( (-4, -2) \) and \( (2, 4) \).
- **(b) \( f(x) > 0 \)**: The function is positive for \( x \in (-3, 3) \).
- **(c) \( f(x) < 0 \)**: The function is negative for \( x \in (-\infty, -3) \cup (3, \infty) \).

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### 35
- **(a) Increasing**: The function is increasing when \( x \) is in \( (0, 3) \).
- **(b) \( f(x) > 0 \)**: The function is positive for \( x \in (1, \infty) \).
- **(c) \( f(x) < 0 \)**: The function is negative for \( x \in (-\infty, 1) \).

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### 36
- **(a) Increasing**: The function is increasing when \( x \) is in \( (-2, 1) \) and \( (3, \infty) \).
- **(b) \( f(x) > 0 \)**: The function is positive for \( x \in (-1, 0) \cup (4, \infty) \).
- **(c) \( f(x) < 0 \)**: The function is negative for \( x \in (-\infty, -1) \cup (0, 4) \).

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Would you like more details or have any specific questions about these explanations?

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### Follow-up Questions
1. How can we identify points where the function changes from increasing to decreasing?
2. What does it mean for a function to be positive or negative on an interval?
3. How can we use critical points to determine increasing or decreasing intervals?
4. Why is it important to know where \( f(x) \) is positive or negative?
5. How does the behavior of the graph around the x-axis help in understanding the function?

### Tip
Look for turning points and x-intercepts on a graph to easily identify changes in behavior, such as increasing or decreasing trends and positive or negative regions.

In Exercises 33-36, describe the x-values for which (a) f is increasing or decreasing, (b) f(x) > 0, and (c) f(x) < 0.

Explore how to determine the increasing, decreasing, positive, and negative intervals for functions shown in Exercises 33-36. This problem includes graphical analysis and function behavior interpretation, ideal for students in Grades 10-12.

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