Let's analyze the graph and solve the four sub-questions.

The graph shows a function $f(x)$, and we need to answer the following questions based on its behavior.

Part A: Solution to the inequality $f(x) > 0$

We need to determine where the function is above the x-axis. From the graph, the function appears to be greater than 0 in two intervals:

• From $x = -6$ to $x = -2$ (the curve is above the x-axis in this range),
• From $x = 2$ to $x \to \infty$.

So the solution is: $f(x) > 0 \quad \text{for} \quad (-6, -2) \cup (2, \infty)$

Part B: One $x$-value for which $f(x) = 0$

From the graph, the function crosses the x-axis at $x = 1$, so one $x$-value for which $f(x) = 0$ is: $x = 1$

Part C: Solution to the inequality $f(x) \geq 0$

This includes where the function is greater than or equal to 0, so we add the points where the function touches the x-axis:

• From $x = -6$ to $x = -2$,
• $x = 1$ (since $f(x) = 0$ here),
• From $x = 2$ to $\infty$.

Thus, the solution is: [ f(x) \geq 0 \quad \text{for} \quad [-6, -2] \cup [1, 1] \cup [2, \infty) ]

Part D: Solution to the inequality $f(x) \leq 0$

This includes where the function is less than or equal to 0. The graph is below the x-axis in the intervals:

• From $x = -2$ to $x = 1$.

So the solution is: $f(x) \leq 0 \quad \text{for} \quad [-2, 1]$

Let me know if you need more details or have any questions! Here are 5 related questions you might consider:

1. How do you determine the intervals where $f(x)$ is positive or negative?
2. How do you handle inequalities when the function touches the x-axis at specific points?
3. What are the differences between solving $f(x) > 0$ and $f(x) \geq 0$?
4. Can a function be positive and negative on the same interval?
5. How can you use the graph of the derivative of a function to solve inequalities like these?

Tip: Always pay attention to the open and closed intervals when solving inequalities involving $f(x)$ because it indicates whether the value at the boundary is included or not!