Question

Find all critical numbers for the function f(x) = (9 - x^2)^(3/5):

A. x = -3, x = 3
B. x = -2, x = 2
C. x = -1, x = 1
D. x = 0

Answer 1

After computing the derivative of the function f(x) = (9 - x^2)^(3/5), we find that the critical numbers occur where the derivative is zero. There is no x value that makes the derivative zero within the domain of f(x), therefore the correct answer is D. x = 0.

Step-by-step explanation:

To find all critical numbers for the function f(x) = (9 - x^2)^(3/5), we must find the values of x where the derivative of the function is zero or undefined. The critical numbers occur when the derivative is zero, since in this case, the function does not have points where the derivative is undefined within the domain of the original function.

First, we compute the derivative of the function using the chain rule:

f'(x) = (3/5)(9 - x^2)^(-2/5)(-2x)

Set f'(x) to zero and solve for x:

0 = (3/5)(9 - x^2)^(-2/5)(-2x)

This implies that -2x = 0, or x = 0. However, x = 0 is not a solution since (9 - x^2)^(-2/5) is not zero or undefined for any x in the domain of the function.

Therefore, there are no critical numbers for the function f(x), and the correct answer is D. x = 0.