9.6 Constrained Optimization

Main Points:
Optimization problem solving is often constrained by certain external things, this is why we need constrained optimization. If the function can be maximized or minimized with the function of a constraint, then that max or min occurs at the point where the constraint function is tangent to the contour of the function being maximized, or at an endpoint of the constraint function. You can also the method of Lagrange Multipliers. The lagrange multiplier represents the change in the optimum value of the function in question when the constraint is increased by one. You can use this fact to construct other equations that relate this fact in relation to each variable (increasing just x or just y) and solve for all three variables with the three equations. The Lagrangian function can also be used, but unfortunately I have no idea how. The best way for me to describe this process is basically finding where the partial derivatives of x, y and Lambda equal 0 (when the function can't increase anymore) . New symbols frighten me.

Challenges:
I don't fully understand the process of solving with Lagrange multipliers. The part I'm having trouble with is how they derive the equations for the Lagrange, and how that relates to x and y of both the original function and the constraint function.

Reflections:
These methods will be very usefull in real world applications, I'm sure. What I need is a person talking through these explanations rather than a book. When in this world of our's are there no constraints?