Question

Which of the extrema below does the function f(x) = x^4 - 2x^2 + x - 2 have?

a. I. Absolute maximum
b. II. Absolute minimum
c. III. Local maximum
d. IV. Local minimum

Answer 1

To determine the extrema of the function f(x) = x^4 - 2x^2 + x - 2, one must compute the first derivative, find critical points, and use the second derivative or first derivative test to classify them. The question's answer requires performing these calculations.

Step-by-step explanation:

The function in question is f(x) = x^4 - 2x^2 + x - 2. To determine the function's extrema, we need to calculate the first derivative, set it equal to zero to find critical points, and then use the second derivative or the first derivative test to classify the type of extrema (local minima, local maxima, absolute minima, and absolute maxima).

First, we find the first derivative of the function: f'(x) = 4x^3 - 4x + 1. Then solve f'(x) = 0 to find critical values. To classify these critical points, we can use the second derivative, f''(x) = 12x^2 - 4, and apply the second derivative test.

We can also explore the behavior of the function as x approaches infinity to see if there is an absolute maximum or minimum.

The student's question cannot be fully answered without performing these computations. Therefore, I would suggest using calculus techniques to categorize the extrema for the given function.