Final answer:
The critical number of the function f(x) = (2x-6)⁴ is found by setting its first derivative equal to zero. The critical number for this function is x = 3.
Step-by-step explanation:
To find the critical number of the function f(x) = (2x-6)⁴, we need to find the values of x where the first derivative f'(x) is equal to zero or undefined. Taking the derivative, we apply the chain rule:
f'(x) = 4(2x - 6)³(2) = 8(2x - 6)³
To find the critical numbers, set the derivative equal to zero:
8(2x - 6)³ = 0
Since the term (2x - 6)³ is never negative, it can only equal zero if the base, 2x - 6, is zero. Hence, we solve for x:
2x - 6 = 0
2x = 6
x = 3
Therefore, the critical number of the function is x = 3.