f(x) = [x(x - 3)]2 is increasing in

To find where the function is increasing, we need to find its derivative and set it to greater than zero.

Step 1: Find the derivative of the function. f'(x) = d/dx [x(x - 3)]^2

Using the chain rule, we get: f'(x) = 2[x(x - 3)] * d/dx [x(x - 3)] = 2[x(x - 3)] * (x - 3 + x) = 2[x(x - 3)] * (2x - 3)

Step 2: Set the derivative greater than zero and solve for x. 2[x(x - 3)] * (2x - 3) > 0

This inequality is satisfied when either of the factors is greater than zero.

For the first factor [x(x - 3)], it is greater than zero when x > 3.

For the second factor (2x - 3), it is greater than zero when x > 3/2.

Therefore, the function f(x) = [x(x - 3)]^2 is increasing when x > 3.