What is a Constraint?

The previous example shows how to find local extrema of a function of two variables, f(x,y) that is free of constraints. Normally, though, there are ``real world'' considerations for optimizing functions which can cause you to reject one extrema in favor of another extrema which may not be the absolute maximum or minimum of the function. For example, with our profit function above, P(x,y), we probably also have a budget function which shows the expense of producing x of product A and y of product B. This cost function, C(x,y), is probably constrained to be less than some value (our budget,) say C₀. Thus, we need to find the production to maximize profit subject to the constraint that $C(x,y) \le C_0$. This means that if our maximum profit occurs at a point (x₀,y₀) which is outside of our constrained region, then we must reject this extremum.