Final answer:
To find the absolute maximum and minimum values of a function in a given interval, we need to analyze the critical points and the endpoints of the interval. The absolute maximum value is 138/27 which occurs at x = 9/2, and the absolute minimum value is -37/4 which occurs at x = -2.
Step-by-step explanation:
To find the absolute maximum and minimum values of a function in a given interval, we need to analyze the critical points and the endpoints of the interval.
First, let's find the critical points by taking the derivative of f(x) and setting it equal to zero: f'(x) = 4 + 42/x^3 = 0
Solving this equation, we get x = ±√10. However, since -2 is not within the interval [-2, 9/2], we can disregard x = -√10 as a critical point.
Next, we evaluate f(x) at the critical point x = √10 and the endpoints -2 and 9/2. The absolute maximum value is the largest of these values, and the absolute minimum value is the smallest.
Calculating f(x) for these points: f(-2) = 4(-2) - (21/(-2)^2) = -8 - 21/4 = -37/4, f(√10) = 4√10 - (21/√10^2) = 4√10 - 21/10, f(9/2) = 4(9/2) - (21/(9/2)^2) = 18 - 84/81 = 138/27
The absolute maximum value is 138/27 which occurs at x = 9/2, and the absolute minimum value is -37/4 which occurs at x = -2.