Final answer:
To find the absolute maximum value of the function f(x) = 4x - 3xln(x) on the interval (0, ∞), we need to examine the critical points and endpoints. The critical point x = e is outside the interval, so we only need to consider the endpoint x = 0. Evaluating f(0), we get the absolute maximum value of 0.
Step-by-step explanation:
The function f(x) = 4x - 3xln(x) is defined on the interval (0, ∞). To find the absolute maximum value of this function on this interval, we need to examine the critical points and endpoints of the interval.
First, let's find the critical points by taking the derivative of f(x). The derivative is f'(x) = 4 - 3ln(x) - 3. Setting this equal to zero and solving for x, we get x = e.
Since the function is only defined on the interval (0, ∞), the critical point x = e is not within this interval. Therefore, we only need to consider the endpoint x = 0. Evaluating f(x) at x = 0, we get f(0) = 0.
So, the absolute maximum value of f(x) on (0, ∞) is 0.