To determine where the function $f(x)$ is increasing or decreasing, we need to look at the slope of the function, typically using the first derivative $f'(x)$. By examining the sign of $f'(x)$, we can determine the behavior of $f(x)$:

1. Increasing: Where $f'(x) > 0$, the function is increasing.
2. Decreasing: Where $f'(x) < 0$, the function is decreasing.

Analysis of the Graph

From the provided graph:

1. The function appears to be increasing from $x = -\infty$ to $x = -1$.
2. At $x = -1$, the slope reaches zero, suggesting a local maximum.
3. The function then decreases from $x = -1$ to $x = 2$.
4. At $x = 2$, the slope again reaches zero, suggesting a local minimum.
5. Finally, the function increases from $x = 2$ to $x = \infty$.

Summary

• Increasing on $(-\infty, -1) \cup (2, \infty)$
• Decreasing on $(-1, 2)$

Would you like further details on identifying increasing and decreasing intervals?

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Here are five related questions for deeper understanding:

1. How do we determine local maxima and minima using the first derivative test?
2. How can we use the second derivative to confirm the concavity of these intervals?
3. What are the critical points of a function, and how do they help in analyzing increasing and decreasing behavior?
4. How would we handle intervals if the function had a discontinuity?
5. How does the nature of an inflection point impact the increasing or decreasing intervals of a function?

Tip: Always look for where the derivative is zero or undefined to find potential intervals of increase or decrease.