The graph of the *derivative* 

*f* ′

 of a function *f* is shown.

The *x* *y*-coordinate plane is given. The curve labeled *y* = *f* ′(*x*) begins on the positive *y*-axis, goes down and right becoming more steep, crosses the *x*-axis at *x* = 1, goes down and right becoming less steep, changes direction at a point below *x* = 2, goes up and right becoming more steep, crosses the *x*-axis at *x* = 3, goes up and right becoming less steep, changes direction at a point above *x* = 4, goes down and right becoming more steep, crosses the *x*-axis at *x* = 5, goes down and right becoming less steep, and ends at a point below *x* = 6.

(a)

On what intervals is *f* increasing? (Enter your answer using interval notation.)

(3,4)
 
 
On what intervals is *f* decreasing? (Enter your answer using interval notation.)

(0,2)∪(4,6)
 
 

To determine where the function \( f(x) \) is increasing or decreasing based on the graph of its derivative \( f'(x) \), we analyze the sign of \( f'(x) \):

- **When \( f'(x) > 0 \)**: \( f(x) \) is increasing.
- **When \( f'(x) < 0 \)**: \( f(x) \) is decreasing.

### Analysis:
1. **Increasing intervals**: The graph of \( f'(x) \) is above the \( x \)-axis on the interval \( (3, 4) \). Hence, \( f(x) \) is **increasing** on \( (3, 4) \).

2. **Decreasing intervals**: The graph of \( f'(x) \) is below the \( x \)-axis on the intervals \( (0, 2) \) and \( (4, 6) \). Therefore, \( f(x) \) is **decreasing** on \( (0, 2) \cup (4, 6) \).

### Final Answer:
- \( f(x) \) is increasing on \( (3, 4) \).
- \( f(x) \) is decreasing on \( (0, 2) \cup (4, 6) \).

Let me know if you need further clarification! Here are 5 related questions to expand your understanding:

1. What happens to \( f(x) \) at points where \( f'(x) = 0 \)?
2. How can the second derivative \( f''(x) \) help analyze the concavity of \( f(x) \)?
3. What are the critical points of \( f(x) \) based on \( f'(x) \)?
4. How would the intervals change if \( f'(x) \) crossed the \( x \)-axis more frequently?
5. What do changes in the steepness of \( f'(x) \) indicate about the behavior of \( f(x) \)?

**Tip**: Always check where \( f'(x) \) changes sign to locate critical points and determine increasing/decreasing behavior!

The graph of the derivative f′ of a function f is shown. The x-y coordinate plane is given. The curve labeled y = f′(x) begins on the positive y-axis, goes down and right becoming more steep, crosses the x-axis at x = 1, goes down and right becoming less steep, changes direction at a point below x = 2, goes up and right becoming more steep, crosses the x-axis at x = 3, goes up and right becoming less steep, changes direction at a point above x = 4, goes down and right becoming more steep, crosses the x-axis at x = 5, goes down and right becoming less steep, and ends at a point below x = 6. On what intervals is f increasing? On what intervals is f decreasing?

Learn how to determine where a function is increasing or decreasing using the graph of its derivative. Analyze intervals based on the sign of f′(x) and apply the First Derivative Test. Suitable for high school calculus students.

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