To determine when the original function is increasing or decreasing, we need to analyze the behavior of its derivative.

Key Concept:

• A function is increasing when its derivative is positive.
• A function is decreasing when its derivative is negative.

From the description, the derivative function crosses the $x$-axis at:

• $x = -1$,
• $x = 0$,
• $x = 1$.

Additionally, there are turning points (changes in direction) at approximately:

• $(-0.58, 0.4)$,
• $(0.58, -0.4)$.

Step-by-Step Analysis:

1. For $x < -1$:

   • The derivative is below the $x$-axis (negative).
   • Thus, the original function is decreasing.

2. For $-1 < x < 0$:

   • The derivative is above the $x$-axis (positive).
   • Thus, the original function is increasing.

3. For $0 < x < 1$:

   • The derivative is below the $x$-axis (negative).
   • Thus, the original function is decreasing.

4. For $x > 1$:

   • The derivative is above the $x$-axis (positive).
   • Thus, the original function is increasing.

Final Answer:

• The function is increasing for $-1 < x < 0$ and $1 < x$.
• The function is decreasing for $x < -1$ and $0 < x < 1$.

Let me know if you'd like more details or have any questions!

Here are 5 related questions:

1. How do you find the critical points of a function given its derivative?
2. What is the relationship between concavity and the second derivative?
3. How does the behavior of the derivative at turning points affect the original function?
4. Can a function be increasing or decreasing if its derivative is zero?
5. How can we determine if the original function has a maximum or minimum at a given point?

Tip: Remember, the sign of the derivative indicates the slope of the original function: positive means increasing, negative means decreasing.