**Category:** Algebra  
**Subtype:** Polynomial Functions  
**Problem Description:** Determine the number of positive integer values of \(m\) for which the function \(y = x^3 - 3(2m + 1)x^2 + (12m + 5)x + 2\) is increasing on the interval \((2, +\infty)\). The options provided indicate the possible number of values in the set \(S\).