Final answer:
The maximum of the function f(x) = 1/x in the interval [-2, -1] occurs at x = -2 with a value of f(-2) = -1/2, and the minimum occurs at x = -1 with a value of f(-1) = -1. As the function is monotonically decreasing in this interval, there are no critical points between the endpoints, and the extrema are found at the endpoints. option B as the correct answer.
Step-by-step explanation:
To find the absolute maximum and minimum of the function f(x) = 1/x on the interval [-2, -1], we must first consider the behavior of the function within the given interval. Since the function is continuous and differentiable over the interval [-2, -1] (excluding x=0 where the function is undefined), we can use calculus to find any critical points and then test the endpoints of the interval to determine the absolute extrema.
To find critical points, we could find the derivative of f(x) and solve for when the derivative is zero or undefined. However, in this interval, the function f(x) is decreasing along the entire interval (as the variable x increases in value from -2 towards -1, the output values decrease, which is common behavior for the function 1/x in the negative domain). Therefore, we do not have any turning points in this interval; the maximum and minimum values must occur at the endpoints.
At x = -2, f(-2) = -1/2 and at x = -1, f(-1) = -1. Comparing these two values, we see that f(-2) > f(-1), which means the maximum occurs at x = -2, and the minimum occurs at x = -1. This identifies option B as the correct answer.