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The exercise you referred to is in Spanish, and you've mentioned that you want help with "Ejercicio 6" (Exercise 6). I will focus on analyzing and solving the sixth exercise from the image.

### Exercise 6 Description (Translated):

**6. For the function \( f(x) = x^3 - 6x^2 + 9x \), determine if it is increasing or decreasing in the given intervals.**

The intervals are typically provided in these types of problems. The function is a polynomial of degree 3, and to determine if the function is increasing or decreasing, you need to:
1. Find the first derivative \( f'(x) \).
2. Determine the sign of \( f'(x) \) in each interval.

#### Solution:

1. **Find the derivative \( f'(x) \):**
   \[
   f(x) = x^3 - 6x^2 + 9x
   \]
   Differentiate with respect to \( x \):
   \[
   f'(x) = 3x^2 - 12x + 9
   \]

2. **Determine the critical points by setting the derivative to zero:**
   \[
   3x^2 - 12x + 9 = 0
   \]
   Divide the equation by 3:
   \[
   x^2 - 4x + 3 = 0
   \]
   Factor the quadratic equation:
   \[
   (x - 3)(x - 1) = 0
   \]
   So, \( x = 3 \) and \( x = 1 \) are the critical points.

3. **Test intervals around the critical points:**

   Consider the intervals determined by the critical points \( x = 1 \) and \( x = 3 \). These intervals are:
   - \( (-\infty, 1) \)
   - \( (1, 3) \)
   - \( (3, \infty) \)

   - For \( x \) in \( (-\infty, 1) \): Pick \( x = 0 \):
     \[
     f'(0) = 3(0)^2 - 12(0) + 9 = 9 \quad (\text{positive, so } f(x) \text{ is increasing})
     \]
   - For \( x \) in \( (1, 3) \): Pick \( x = 2 \):
     \[
     f'(2) = 3(2)^2 - 12(2) + 9 = 12 - 24 + 9 = -3 \quad (\text{negative, so } f(x) \text{ is decreasing})
     \]
   - For \( x \) in \( (3, \infty) \): Pick \( x = 4 \):
     \[
     f'(4) = 3(4)^2 - 12(4) + 9 = 48 - 48 + 9 = 9 \quad (\text{positive, so } f(x) \text{ is increasing})
     \]

4. **Conclusion:**
   - The function is **increasing** on \( (-\infty, 1) \) and \( (3, \infty) \).
   - The function is **decreasing** on \( (1, 3) \).

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### Related Questions:

1. How do you find the local maxima and minima of a cubic function?
2. What is the significance of the first derivative test in determining the behavior of a function?
3. How can you use the second derivative to confirm concavity and points of inflection?
4. What are the general steps to sketching the graph of a polynomial function?
5. How does the sign of the derivative affect the shape of the graph?

**Tip:** Remember that finding the critical points by setting the first derivative to zero is key to understanding the behavior of the function across different intervals.

Learn how to determine if the function f(x) = x^3 - 6x^2 + 9x is increasing or decreasing over different intervals. This problem involves calculus concepts such as derivatives and the first derivative test, applicable to polynomial functions.

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