Final answer:
To find the critical numbers for the function f(x) = 3x⁴ - 4x³ - 12x², we differentiate it, set the derivative equal to zero, and solve for x. The critical numbers are x = 0, x = 2, and x = -1.
Step-by-step explanation:
To find all critical numbers for the function f(x) = 3x⁴ - 4x³ - 12x², we need to find the values of x where the derivative of the function, f'(x), is equal to zero or undefined. The first step is to calculate the derivative:
f'(x) = 12x³ - 12x² - 24x.
Next, we set the derivative equal to zero and solve for x:
0 = 12x³ - 12x² - 24x.
We can factor out a 12x from each term:
0 = 12x(x² - x - 2).
Further factoring yields:
0 = 12x(x - 2)(x + 1).
Therefore, the critical numbers are x = 0, x = 2, and x = -1.