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.