Section 9.6

Main Points: Constrained optimization is used when certain limits are placed on a problem, such as a budget. To maximize something subject to contraints using a graph, the point on the contour line completely exhausts the budget, point below the line gives possible values of x and y that will not use all of the budget, and a point about the contour line cannot be afforded. The maximum occurs at a point P where the budget constraint is tangent to the production contour. Lagrange multipliers solves three equations: partial derivative of f with respect to x (x,y) = lambda g(sub x) of (x,y). The second equation is the same except with respect to y. Finally g(x,y)= c (or some constant). lambda is the Lagrange multiplier. If f has a constrained max or min, then it occurs at one of the Lagrange solutions or at the endpoint of the constraint. Lambda is approximately equal to the change in optimum value of f when the value of the constraint is increased by one unit. Also, lambda represents the rate of change of optimum values of f as the constraint increases. Finally, the Lagrangian function allows a person solving a constraint optimization problem the opportunity to use 2 steps: first, write the Lagrangian function and then find the critical points of that function.

Challenges: I'm unsure about how to use the Lagrangian function. I understand that the answers from the equation are the critical points, but I don't understand how to get an answer from it, if that makes sense?

Reflections: This is really helpful in getting the most for your money. Because companies and businesses want to be able to make more money than they spend, finding the optimization under constraints is a really important concept.