The graph of the derivative of a function is given.

The *x* *y*-coordinate plane is given. The curve enters the window in the third quadrant, goes up and right, crosses the *x*-axis at *x* = −1, changes direction at the approximate point (−0.58, 0.4), goes down and right, crosses the *x*-axis at the origin, changes direction at the approximate point (0.58, −0.4), goes up and right, crosses the *x*-axis at *x* = 1, and exits the window in the first quadrant.

For which *x* is the (original) function increasing? For which *x* is the (original) function decreasing?

The function is increasing for −1 < *x* <  and *x*  ? < >  , and decreasing for *x*  ? < >  and 0 < *x* <  .

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.

The graph of the derivative of a function is given. The curve enters the window in the third quadrant, goes up and right, crosses the x-axis at x = −1, changes direction at the approximate point (−0.58, 0.4), goes down and right, crosses the x-axis at the origin, changes direction at the approximate point (0.58, −0.4), goes up and right, crosses the x-axis at x = 1, and exits the window in the first quadrant. For which x is the (original) function increasing? For which x is the (original) function decreasing? The function is increasing for −1 < x < and x ? < > , and decreasing for x ? < > and 0 < x < .

Understand how to determine where a function is increasing or decreasing by analyzing its derivative graph. The problem explores key concepts in calculus, such as the first derivative test and the behavior of functions based on the sign of the derivative. Suitable for students in Grades 11-12.

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