Final answer:
To find the absolute minimum and maximum values of f(x) = x - ln(9x) on the interval [1/2, 2], we need to find the critical points and endpoints. The critical point is x = 3/2 and the function values at the critical point and endpoints are -0.81, 0.75, and 0.39 respectively. Therefore, the absolute minimum value is approximately -0.81 and the absolute maximum value is approximately 0.75.
Step-by-step explanation:
To find the absolute minimum and maximum values of the function f(x) = x - ln(9x) on the interval [1/2, 2], we need to find the critical points and endpoints of the interval.
First, let's find the critical points by setting the derivative f'(x) equal to zero and solving for x:
f'(x) = 1 - 1/(9x) = 0
After solving, we find x = 3/2 as the critical point.
Next, we evaluate the function at the critical point and endpoints to find the absolute minimum and maximum values:
f(1/2) = 1/2 - ln(9/4) ≈ -0.81
f(3/2) = 3/2 - ln(27/18) ≈ 0.75
f(2) = 2 - ln(18) ≈ 0.39
Therefore, the absolute minimum value is approximately -0.81 and the absolute maximum value is approximately 0.75.