Definition: A subtility is a term used in mathematics to describe an object or mathematical function that behaves like a constant, but whose value changes with respect to another variable. Subutilities are often used in optimization problems where one wants to find the most efficient solution by minimizing some objective function. For example, consider the function f(x) = 2x^3 - 3x^2 + x + 1. This function is defined for all real numbers x and can be minimized using calculus techniques,