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.

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 $x \in (-\infty, 1) \cup (5, \infty)$.

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 $x \in (-\infty, -3) \cup (3, \infty)$.

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 $x \in (-\infty, 1)$.

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 $x \in (-\infty, -1) \cup (0, 4)$.

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

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.