Final answer:
To find the absolute extremum of g(x) = -(x - 3/e^x) on the interval (0,6), one must calculate the derivative, find the critical points, evaluate the endpoints, and compare the values to determine the absolute maximum and minimum.
Step-by-step explanation:
To find an absolute extremum of the function g(x) = -(x - 3/ex) on the interval (0,6), you need to follow a few steps. First, find the derivative of the function, which will help us locate any critical points where the extremum might occur. Then, we check the endpoints of the interval since absolution extrema can occur there.
1. Find the first derivative g'(x), which is g'(x) = -1 + (3/ex)*x.
2. Solve g'(x) = 0 for critical points within the interval (0,6).
3. Evaluate the function g(x) at the critical points and the endpoints x=0 and x=6.
4. Compare these values to determine the absolute maximum and minimum of g(x) on (0,6).
By using these steps, you can conclude which are the absolute extremum values of the given function in the specified interval.