If x<l, then f'(*) = 6()>0 so f; If1<x< 2, then f(x) = 6(+)
(<0 and sof: If x> 2, then f' (x) = 6(+) (+) > 0 andso ↑. ,
Thusf is increasing on (-, 1) and (2, 0) and is decreasing on (1
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It looks like the text you've provided is a bit scrambled, but I can help clarify the mathematical reasoning for determining the behavior of a function \( f(x) \) based on the sign of its derivative \( f'(x) \).

Here is a clearer interpretation of the provided information:

1. **When \( x < 1 \):**  
   \( f'(x) = 6(\ldots) > 0 \)  
   This indicates that the derivative of \( f(x) \) is positive when \( x \) is less than 1. Therefore, the function \( f(x) \) is **increasing** on the interval \( (-\infty, 1) \).

2. **When \( 1 < x < 2 \):**  
   \( f'(x) = 6(\ldots) < 0 \)  
   This shows that the derivative is negative for \( x \) in the interval \( (1, 2) \). Consequently, the function \( f(x) \) is **decreasing** on this interval.

3. **When \( x > 2 \):**  
   \( f'(x) = 6(\ldots) > 0 \)  
   The derivative becomes positive again for \( x > 2 \), meaning that \( f(x) \) is **increasing** on the interval \( (2, \infty) \).

### Summary:
- \( f(x) \) is **increasing** on the intervals \( (-\infty, 1) \) and \( (2, \infty) \).
- \( f(x) \) is **decreasing** on the interval \( (1, 2) \).

Would you like me to go into further details or explain anything else?

Here are five related questions that might interest you:
1. How do you determine the critical points of a function using its derivative?
2. What is the significance of the second derivative in analyzing the behavior of a function?
3. How do inflection points relate to the concavity of a function?
4. Can you explain how to sketch a graph of a function based on its first and second derivatives?
5. What are some common applications of increasing and decreasing functions in real-life scenarios?

**Tip:** Understanding the signs of the first derivative helps you determine where a function is increasing or decreasing, which is crucial for sketching its graph and understanding its behavior.

If x<l, then f'(*) = 6()>0 so f; If1<x< 2, then f(x) = 6(+) (<0 and sof: If x> 2, then f' (x) = 6(+) (+) > 0 andso ↑. Thusf is increasing on (-, 1) and (2, 0) and is decreasing on (1.

Learn how to determine where a function is increasing or decreasing by analyzing the sign of its first derivative. This detailed solution covers key calculus concepts like derivatives and the first derivative test. Suitable for students in Grades 11-12.

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