To find the critical numbers of the function f(x) = x^2(x-3)^2, we need to find the values of x where the derivative of f(x) is zero or undefined.
First, we find the derivative of f(x) using the product rule:
f'(x) = 2x(x-3)^2 + x^2(2(x-3))
Simplifying, we get:
f'(x) = 2x(x-3)(2x-6+x) = 2x(x-3)(3x-6)
Setting f'(x) = 0, we get:
2x(x-3)(3x-6) = 0
This equation is zero when x = 0, x = 3, and x = 2.
Now we need to check if these values make the derivative undefined. None of these values make the derivative undefined.
Therefore, the critical numbers of the function f(x) = x^2(x-3)^2 are x = 0, x = 3, and x = 2.