Final answer:
The absolute maximum value of the function f(x) = x - 4ln(x) is -1.544 and it occurs at x = 4. The absolute minimum value is -5.294 and it occurs at x = 1/4.
Step-by-step explanation:
In the given function f(x) = x - 4ln(x), the absolute maximum value occurs when the derivative of the function is equal to zero. To find the derivative, we use the product rule and the chain rule:
f'(x) = 1 - \frac{4}{x}
Setting f'(x) equal to zero gives us:
1 - \frac{4}{x} = 0
Solving for x, we get x = 4. The absolute maximum value of the function is f(4) = 4 - 4ln(4) = 4 - 4(1.386) = 4 - 5.544 = -1.544. So, the absolute maximum value is -1.544 and it occurs at x = 4.
To find the absolute minimum value, we need to evaluate the function at the endpoints of the given interval, which are x = 1/4 and x = 8:
f(1/4) = \frac{1}{4} - 4ln\left(\frac{1}{4}\right) = \frac{1}{4} + 4ln\left(4\right) = \frac{1}{4} - 4(1.386) = \frac{1}{4} - 5.544 = -5.294
f(8) = 8 - 4ln(8) = 8 - 4(2.079) = 8 - 8.316 = -0.316
Therefore, the absolute minimum value is -5.294 and it occurs at x = 1/4.