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f (x) = x^3 − 6x^2 + 9x + 1

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.

Learn how to determine intervals of increase and decrease for the function f(x) = x^3 - 6x^2 + 9x + 1 using derivatives and critical points. This problem covers key calculus concepts, including derivatives and their applications in analyzing functions. Suitable for advanced high school students and beyond.

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