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Find critical number f(x)= (2x-6)⁴

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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.

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