Final answer:
The absolute maximum value of g(x) on the interval [1, 4] is approximately 0.278.
Step-by-step explanation:
To find the absolute maximum value of the function g(x) = (lnx)/(x) on the interval [1, 4], we need to evaluate the function at the critical points and endpoints in that interval.
We start by finding the derivative of g(x) using the quotient rule: g'(x) = (x(1/x) - ln(x)(1))/(x^2) = (1 - ln(x))/(x^2).
Next, we set g'(x) equal to zero to find the critical point: (1 - ln(x))/(x^2) = 0.
Solving this equation, we get x = e, where e is the base of the natural logarithm.
Next, we evaluate g(x) at the critical point and the endpoints of the interval: g(1) = 0, g(e) ≈ 0.278, and g(4) ≈ 0.087.
Comparing these values, we find that the absolute maximum value of g(x) on the interval [1, 4] is approximately 0.278.