Question

Find the critical numbers of the function f(x) = x⁸(x − 3)⁷ a) x=0 and x=3

b) x=0,x=3 and x=8
c) x=0 only
d) x=3 only

Answer 1

The correct answer is option a. The critical numbers of the function f(x) = x⁸(x - 3)⁷ are found by taking the derivative of the function and setting it equal to zero, leading to the critical numbers x=0 and x=3. This corresponds to option a.

Step-by-step explanation:

To find the critical numbers of the function f(x) = x⁸(x − 3)⁷, we need to determine the values of x for which the first derivative of the function is equal to zero or undefined.

First, take the derivative of the function using the product rule and the power rule:

• f'(x) = d/dx [x⁸(x − 3)⁷]
• f'(x) = x⁸ ∙ 7(x − 3)⁶ + 8x⁷(x − 3)⁷

Now, we set the derivative equal to zero to find the critical points:

• 0 = x⁸ ∙ 7(x − 3)⁶ + 8x⁷(x − 3)⁷
• 0 = x⁷ [7x(x − 3)⁶ + 8(x − 3)⁷]

Note that x⁷ is a common factor, which means one critical number is x = 0. The remaining factor gives us:

• 0 = 7x(x − 3)⁶ + 8(x − 3)⁷
• 0 = (x − 3)⁶ [7x + 8(x − 3)]

Since (x − 3)⁶ cannot be zero unless x = 3, another critical number is x = 3. Therefore, the critical numbers of the function are x=0 and x=3, which corresponds to option a.