130k views
2 votes
Find the absolute minimum and absolute maximum values of f on the given interval.

f(x) = x − ln
(9x) on [1/2, 2]
A Minimum =____
B Maximum = _______

User Remon Amin
by
7.3k points

1 Answer

2 votes

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.

User Marek Lisiecki
by
8.4k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.