To determine the intervals where the function $f(x) = x^3 - 6x^2 + 9x + 1$ is increasing or decreasing, follow these steps:

1. Find the first derivative of the function

The first derivative of $f(x)$ will help us find the critical points, which are the points where the function's slope changes.

$f'(x) = \frac{d}{dx}\left(x^3 - 6x^2 + 9x + 1\right)$

Differentiating term by term:

$f'(x) = 3x^2 - 12x + 9$

2. Find the critical points

Set the first derivative equal to zero to find the critical points:

$3x^2 - 12x + 9 = 0$

This is a quadratic equation. We can solve it using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = 3$, $b = -12$, and $c = 9$:

$x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(3)(9)}}{2(3)}$

Simplifying:

$x = \frac{12 \pm \sqrt{144 - 108}}{6}$ $x = \frac{12 \pm \sqrt{36}}{6}$ $x = \frac{12 \pm 6}{6}$

So, the critical points are:

$x = \frac{18}{6} = 3 \quad \text{and} \quad x = \frac{6}{6} = 1$

3. Determine the intervals of increase and decrease

To find where the function is increasing or decreasing, examine the sign of $f'(x)$ on the intervals determined by the critical points $x = 1$ and $x = 3$.

• For $x < 1$: Choose a test point like $x = 0$: f'(0) = 3(0)^2 - 12(0) + 9 = 9 \quad (\text{positive, so f(x) is increasing})

• For $1 < x < 3$: Choose a test point like $x = 2$: f'(2) = 3(2)^2 - 12(2) + 9 = 12 - 24 + 9 = -3 \quad (\text{negative, so f(x) is decreasing})

• For $x > 3$: Choose a test point like $x = 4$: f'(4) = 3(4)^2 - 12(4) + 9 = 48 - 48 + 9 = 9 \quad (\text{positive, so f(x) is increasing})

Conclusion

• The function $f(x)$ is increasing on the intervals $(-\infty, 1)$ and $(3, \infty)$.
• The function $f(x)$ is decreasing on the interval $(1, 3)$.

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:

1. How do you determine concavity and points of inflection of a function?
2. What is the significance of the second derivative in analyzing a function?
3. How can you use critical points to find local maxima and minima?
4. How do you find the global maximum and minimum of a function on a closed interval?
5. What are the possible behaviors of a function around critical points?

Tip: Always check the behavior of the first derivative on intervals around the critical points to determine where the function is increasing or decreasing.