Final answer:
To find the absolute maximum and minimum values of the function f(x) = x^(-2) * ln(x) over the interval [1/2, 7], we need to find the critical points and the endpoints of the interval.
Step-by-step explanation:
To find the absolute maximum and minimum values of the function f(x) = x^(-2) * ln(x) over the interval [1/2, 7], we need to find the critical points and the endpoints of the interval.
First, let's find the critical points. We find the derivative of f(x) and set it equal to zero: f'(x) = -2x^(-3) * ln(x) + x^(-3) / x = 0.
Simplifying, we get -2ln(x) + 1 = 0. Solving for x, we find x = e^(1/2).
Next, we evaluate f(x) at the critical point and the endpoints of the interval to determine the maximum and minimum values. Plugging in x = 1/2, 7, and e^(1/2), we find f(1/2) = -2ln(1/2), f(7) = -2ln(7), and f(e^(1/2)). The largest value among these is the absolute maximum, and the smallest value is the absolute minimum.