Let's analyze the given problem to determine the intervals where the function is increasing, decreasing, or constant.

The function is:

$f(x) = x^3 - 9x^2 + 3$

To determine the intervals of increase, decrease, or constant behavior, we will:

1. Find the derivative: This will help us locate critical points and determine where the function is increasing or decreasing.

$f'(x) = 3x^2 - 18x$

2. Factor the derivative: $f'(x) = 3x(x - 6)$

3. Find critical points: Set the derivative equal to zero to find the critical points. $3x(x - 6) = 0$ So, the critical points are: $x = 0 \quad \text{and} \quad x = 6$

4. Test intervals: Now, test the intervals around the critical points to determine if the function is increasing or decreasing in each interval.

   • For $x < 0$ (e.g., $x = -1$): $f'(-1) = 3(-1)(-1 - 6) = 3(-1)(-7) = 21 > 0$ (increasing)
   • For $0 < x < 6$ (e.g., $x = 1$): $f'(1) = 3(1)(1 - 6) = 3(1)(-5) = -15 < 0$ (decreasing)
   • For $x > 6$ (e.g., $x = 7$): $f'(7) = 3(7)(7 - 6) = 3(7)(1) = 21 > 0$ (increasing)

5. Conclusion:

   • The function is increasing on the intervals: $(-\infty, 0) \cup (6, \infty)$
   • The function is decreasing on the interval: $(0, 6)$
   • The function is never constant.

Would you like further details on this analysis or have any questions?

Here are 5 relative questions:

1. How does the derivative test help in identifying increasing and decreasing intervals?
2. What is the significance of critical points in this context?
3. Can a function be constant if its derivative is zero at isolated points?
4. What is the behavior of cubic functions in general?
5. How would the graph look if we analyzed higher-order terms?

Tip: Always confirm the nature of the critical points by testing intervals around them to ensure the correct classification of increasing and decreasing regions.