Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
ƒ(x) = x³ ln x on (0, ∞)

Verified step by step guidance

1. First, find the derivative of the function ƒ(x) = x³ ln x using the product rule, since it is a product of two functions: x³ and ln x.
2. Set the derivative equal to zero to find the critical points: ƒ'(x) = 0. This will help identify where the function's slope is zero, indicating potential maxima or minima.
3. Determine the domain of the function and the critical points found in the previous step to ensure they lie within the interval (0, ∞).
4. Evaluate the function ƒ(x) at the critical points and also consider the behavior of the function as x approaches the boundaries of the interval (0 and ∞) to find the absolute maximum and minimum values.
5. Compare the values obtained from the critical points and the limits at the boundaries to identify the absolute maximum and minimum values of the function on the interval (0, ∞).