Find the absolute maximum value and the absolute minimum value, if any, of the given function.
A)Absolute maximum value: 199; absolute minimum value: - 1
B)Absolute maximum value: 3; absolute minimum value: - 1
C)Absolute maximum value: 3; absolute minimum value: - 109
D)Absolute maximum value: 199; absolute minimum value: - 109
E)No absolute extrema

Question

Find the absolute maximum value and the absolute minimum value, if any, of the given function. 
A)Absolute maximum value: 199; absolute minimum value: - 1 
B)Absolute maximum value: 3; absolute minimum value: - 1 
C)Absolute maximum value: 3; absolute minimum value: - 109 
D)Absolute maximum value: 199; absolute minimum value: - 109 
E)No absolute extrema

Quizplus · Accepted Answer

Since the domain is not specified, we assume that it is the set of all real numbers. To find the absolute extrema, we need to check the critical points and the endpoints of the domain. The derivative of the function is f'(x) = -2x + 2, which is zero at x = 1. Therefore, we need to check the function at x = 1 and at the endpoints of the domain. The function is unbounded as x approaches infinity or negative infinity, so we do not need to consider the endpoints.

When x = 1, we have f(1) = (1 - 1)^3 - 109(1 - 1)^2 - 1 = -1.

Since there are no other critical points, the absolute maximum must occur either at x = 1 or at one of the endpoints of the domain, and the absolute minimum must occur either at x = 1 or at negative infinity or infinity. Since the function is unbounded at infinity, the absolute maximum occurs at x = 1, with a value of f(1) = -1. The absolute minimum occurs at negative infinity, with a value of -199, since the function is decreasing for x < 1 and unbounded as x approaches negative infinity. Therefore, the answer is D.

Find the Absolute Maximum Value and the Absolute Minimum Value