Question

Find the critical numbers of the function f(x) = x^(1/4) - x^(3/4):

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

Answer 1

The critical numbers of the function f(x) = x^(1/4) - x^(3/4) are calculated by finding where the derivative of the function equals zero. However, the correct critical numbers (0 and 9) are not listed among the provided options, indicating a possible error in the question.

Step-by-step explanation:

To find the critical numbers of the function f(x) = x1/4 - x3/4, we need to differentiate the function and find where the derivative is zero or undefined. The derivative of f(x) is given by:

f'(x) = 1/4 x-3/4 - 3/4 x-1/4

To find the critical numbers, we set the derivative equal to zero:

0 = 1/4 x-3/4 - 3/4 x-1/4

Multiplying through by 4x3/4 to clear the fractions, we get:

0 = x - 3x1/2

We can now factor out x1/2:

0 = x1/2(x1/2 - 3)

This gives us two solutions for x:

• x1/2 = 0
• x1/2 =0

Solving these, we get x = 0 and x = 9 (since x = 32). There was an oversight in the provided options as the correct critical numbers are 0 and 9, which is not listed, therefore, the correct answer is not given in the options.