Final answer:
The question seeks the minimum labor cost based on a given function, but due to unclear variable definitions and inconsistent context, it's not possible to provide an accurate answer without additional information.
Step-by-step explanation:
The student asked to find the minimum labor cost given the function (X, Y) = (3/2)x² + y² - 2x - 6y - 2xy + 121. This appears to be a problem of finding the minimum value of a function, which is often solved using calculus or by completing the square in each of the variables. However, the provided information seems to be confused, as the function given is not defined properly with a clear indication of what X and Y are. We could treat this as a two-variable optimization problem, yet, the problem seems to need more clarification.
Without a clear definition of the variables, especially given the context provided mentioning widgets and workers, we cannot straightforwardly apply the function to find the minimum cost. More information or proper context is necessary to proceed correctly with this question.