As the value of c c increases, the curve shifts to the right. In Figure 4.59, the value c c represents different profit levels (i.e., values of the function f ). A graph of various level curves of the function f ( x, y ) f ( x, y ) follows.įigure 4.59 Graph of level curves of the function f ( x, y ) = 48 x + 96 y − x 2 − 2 x y − 9 y 2 f ( x, y ) = 48 x + 96 y − x 2 − 2 x y − 9 y 2 corresponding to c = 150, 250, 350, and 400. This constraint, when combined with the profit function f ( x, y ) = 48 x + 96 y − x 2 − 2 x y − 9 y 2, f ( x, y ) = 48 x + 96 y − x 2 − 2 x y − 9 y 2, is an example of an optimization problem, and the function f ( x, y ) f ( x, y ) is called the objective function. The goal is, still, to maximize profit, but now there is a different type of constraint on the values of x x and y. Suppose these were combined into a budgetary constraint, such as 20 x + 4 y ≤ 216, 20 x + 4 y ≤ 216, that took into account the cost of producing the golf balls and the number of advertising hours purchased per month. In that example, the constraints involved a maximum number of golf balls that could be produced and sold in 1 1 month ( x ), ( x ), and a maximum number of advertising hours that could be purchased per month ( y ). Lagrange MultipliersĮxample 4.41 was an applied situation involving maximizing a profit function, subject to certain constraints.

In this section, we examine one of the more common and useful methods for solving optimization problems with constraints. However, techniques for dealing with multiple variables allow us to solve more varied optimization problems for which we need to deal with additional conditions or constraints.

Solving optimization problems for functions of two or more variables can be similar to solving such problems in single-variable calculus. 4.8.2 Use the method of Lagrange multipliers to solve optimization problems with two constraints.4.8.1 Use the method of Lagrange multipliers to solve optimization problems with one constraint.

Cobb douglas function