To analyze the graph and determine the intervals where the function $f(x)$ is increasing, decreasing, and constant, we'll look at the behavior of the graph:

1. Increasing Intervals:

• The function is increasing where the graph moves upwards as you move from left to right.
• Based on the graph:
  • From $x = -2$ to $x = -1$
  • From $x = 2$ onwards to the right (towards infinity).

2. Decreasing Intervals:

• The function is decreasing where the graph moves downwards as you move from left to right.
• Based on the graph:
  • From $x = -\infty$ to $x = -2$
  • From $x = -1$ to $x = 2$.

3. Constant Intervals:

• The function is constant where the graph is a horizontal line (no change in y-value as x changes).
• Based on the graph:
  • There are no intervals where the function is constant.

Summary:

• Increasing: $(-2, -1)$ and $(2, \infty)$
• Decreasing: $(-\infty, -2)$ and $(-1, 2)$
• Constant: None

Would you like further details on this analysis, or do you have any other questions? Here are some related questions to consider:

1. How can you determine the local maxima and minima from a graph?
2. What are the effects of critical points on the shape of a graph?
3. How do you identify points of inflection on a graph?
4. What is the significance of the first derivative in determining increasing and decreasing intervals?
5. How can you use the second derivative to analyze concavity and points of inflection?

Tip: Remember, the derivative of a function gives us the slope of the tangent line at any point on the graph, which is crucial for understanding where the function is increasing or decreasing.