Answer: To find the critical numbers of the function, we need to find the values of x where the derivative of the function is equal to zero or does not exist.
Given function: f(x) = 3x^4 + 4x^3 - 12
Step 1: Find the derivative of the function f(x) with respect to x (f'(x)):
f'(x) = d/dx(3x^4) + d/dx(4x^3) - d/dx(12)
f'(x) = 12x^3 + 12x^2
Step 2: Set the derivative f'(x) equal to zero and solve for x to find the critical points:
12x^3 + 12x^2 = 0
Step 3: Factor out the common term 12x^2 from the equation:
12x^2(x + 1) = 0
Step 4: Set each factor equal to zero and solve for x:
12x^2 = 0
x^2 = 0
x = 0
x + 1 = 0
x = -1
So, the critical numbers of the function f(x) = 3x^4 + 4x^3 - 12 are x = 0 and x = -1.