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.