Optimization

Optimizing with a function of multi variables is similar to optimizing with a function of single variable. Instead of using axis, we use planes; instead of finding the maximum area, we find the maximum volume. The process is the same -- we scale down from three variables to two variables.

The Steps to Optimization are as follows:
1) Read and decode the problem
2) Determine the objective function - The thing you are trying to maximize/minimize
3) Identify constraints and relationships
4) Optimize (calculus!!)
5) Answer the question

In the example we did in class, we were asked to find the maximum volume of a prism given that a rectangular prism has a vertex on the origin, faces on the xy-, yz-, xz- plane, and on the plane 5x+4y+2z=20.

The objective function is v(x,y,z) = xyz; by using the plane given we can change 5x+4y+2z=20 into z = 10-5/2x-2y. We can eliminate one variable, leaving us with the function v(x, y) = xy(10-5/2x-2y).

Next, we find the partials in order to find the critical points
Vx = 10y-5xy-2y^2 Vxx = -5y = -25/3
Vy = 10x-5/2x^2-4xy Vyy = -4x = -16/3 Vxy = 10-5x-4y = 100/9

Since d = (-25/3)(-16/3) - 100/9 > 0, (4/3, 5/3) is at a relative maximum.

By solving a system of equations we can determine that the critical point is (4/3, 5/3). Using what we learned about critical points, we can say that there is a relative maximum which occurs at this point.

Be sure to do not only the textbook questions but also the Utexas assignment, and happy optimizing!