Find the maximum value and the corresponding x -value at which the maximum occurs for the function .
A) Maximum value of 12 occurs when
B) Maximum value of 28 occurs when 12
C) Maximum value of 4 occurs when 0
D) Maximum value of 12 occurs when 28
E) Maximum value of 28 occurs when

Question

Find the maximum value and the corresponding x -value at which the maximum occurs for the function  .
A) Maximum value of 12 occurs when 
B) Maximum value of 28 occurs when 12 
C) Maximum value of 4 occurs when 0 
D) Maximum value of 12 occurs when 28 
E) Maximum value of 28 occurs when

Quizplus · Accepted Answer

Taking the derivative of the given function with respect to x and setting it equal to 0 to find critical points:

f'(x) = 264x^(3/4)-124x^(1/4)
0 = 4x^(1/4)(66x^(1/2)-31)
x = (31/66)^2 = 0.475

We now need to check whether this point corresponds to a maximum or a minimum. Taking the second derivative:
f''(x) = 264(3/4)x^(-1/4)-31/4x^(-3/4)

f''(0.475) = 231.33

Since the second derivative is positive, the critical point at x = 0.475 corresponds to a local minimum. Thus, the maximum value of the function occurs at one of the endpoints of the interval [0,1] which are x = 0 and x = 1.

f(0) = 0 and f(1) = 28, so the maximum value of 28 occurs at x = 1.

Therefore, the answer is choice B.

Find the Maximum Value and the Corresponding X -Value at Which