Final answer:
The critical numbers of the function f(x) = 2x³ - 24x² + 72x - 6 are found by setting its first derivative to zero. Factoring the resulting quadratic reveals the critical numbers: the smaller being x = 2 and the larger being x = 6.
Step-by-step explanation:
To find the critical numbers of the function f(x) = 2x³ - 24x² + 72x - 6, we first need to locate the values of x where the first derivative f'(x) is equal to zero or does not exist. Calculating the first derivative, we get:
f'(x) = 6x² - 48x + 72.
We then set the derivative equal to zero to find the critical points:
6x² - 48x + 72 = 0.
Factoring out a 6, we get:
x² - 8x + 12 = 0.
Now we can factor the quadratic:
(x - 2)(x - 6) = 0.
Thus, the critical numbers are x = 2 and x = 6, where x = 2 is the smaller one, and x = 6 is the larger one.
Therefore, the smaller critical number is x = 2 and the larger critical number is x = 6.