Final answer:
The term 'partition numbers' seems to be used instead of 'critical points.' The critical point for the function f(x) = 12x - 8ln(x) is found by setting the first derivative equal to zero, which gives x = 2/3 within the domain x > 0.
Step-by-step explanation:
The term "partition numbers" in the context of the function f'(x) typically means finding the critical points which are values of x where the first derivative is zero or undefined, which can potentially partition the domain of the function into intervals where the function increases or decreases. However, the term "partition numbers" could be a typo, and the student may be asking for critical points. To find critical points of the function f(x) = 12x - 8ln(x), we first find the derivative f'(x). The derivative using basic rules of differentiation is f'(x) = 12 - (8/x). Setting the derivative equal to zero gives us the equation 12 - (8/x) = 0. Solving for x gives us the critical point x = 2/3. Since x > 0 is part of the original function's domain, this critical point is within the domain and hence a partition number, if that refers to critical points.