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The function in the image is \( f(x) = 3x^3 - 9x^2 + 24x - 19 \). To find the intervals where the function is increasing and decreasing, we need to compute its derivative, find the critical points, and analyze the sign of the derivative.

### Steps:

1. **Compute the Derivative**:
   \[
   f'(x) = 9x^2 - 18x + 24
   \]

2. **Find the Critical Points**:
   Set \( f'(x) = 0 \) and solve for \( x \):
   \[
   9x^2 - 18x + 24 = 0
   \]
   We can simplify and solve this quadratic equation.

3. **Determine Intervals of Increase and Decrease**:
   Use the critical points to test the intervals on the derivative to determine where \( f(x) \) is increasing (where \( f'(x) > 0 \)) and decreasing (where \( f'(x) < 0 \)).

Let's solve the quadratic equation to find the critical points and then determine the intervals.

Identify the interval(s) where the function f(x) = 3x^3 - 9x^2 + 24x - 19 is increasing and decreasing.

Learn how to find the intervals where the function f(x) = 3x^3 - 9x^2 + 24x - 19 is increasing and decreasing by using derivatives. This problem involves calculating the first derivative and applying the quadratic formula to find critical points, suitable for students in Grades 11-12 or college level.

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