{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

notes-unconstrained-max - OPMT 5701 Two Variable...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
OPMT 5701 Two Variable Optimization Using Calculus For Maximization Problems One Variable Case If we have the following function y = 10 x x 2 we have an example of a dome shaped function. To fi nd the maximum of the dome, we simply need to fi nd the point where the slope of the dome is zero, or dy dx = 10 2 x = 0 10 = 2 x x = 5 and y = 25 Two Variable Case Suppose we want to maximize the following function z = f ( x, y ) = 10 x + 10 y + xy x 2 y 2 Note that there are two unknowns that must be solved for: x and y . This function is an example of a three-dimensional dome. (i.e. the roof of BC Place ) To solve this maximization problem we use partial derivatives. We take a partial derivative for each of the unknown choice variables and set them equal to zero z x = f x = 10 + y 2 x = 0 The slope in the ”x” direction = 0 z y = f y = 10 + x 2 y = 0 The slope in the ”y” direction = 0 This gives us a set of equations, one equation for each of the unknown variables. When you have the same number of independent equations as unknowns, you can solve for each of the unknowns. rewrite each equation as y = 2 x 10 x = 2 y 10 substitute one into the other x = 2(2 x 10) 10 x = 4 x 30 3 x = 30 1
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
x = 10 similarly, y = 10 REMEMBER: To maximize (minimize) a function of many variables you use the technique of partial di ff erentiation. This produces a set of equations, one equation for each of the unknowns. You then solve the set of equations simulaneously to derive solutions for each of the unknowns. Second order Conditions (second derivative Test) To test for a maximum or minimum we need to check the second partial derivatives. Since we have two fi rst partial derivative equations ( f x , f y ) and two variable in each equation, we will get four second partials ( f xx , f yy , f xy , f yx ) Using our original fi rst order equations and taking the partial derivatives for each of them (a second time) yields: f x = 10 + y 2 x = 0 f y = 10 + x 2 y = 0 f xx = 2 f yy = 2 f xy = 1 f yx = 1 The two partials, f xx , and f yy are the direct e ff ects of of a small change in x and y on the respective slopes in in the x and y direction. The partials, f xy and f yx are the indirect e ff ects, or the cross e ff ects of one variable on the slope in the other variable’s direction. For both Maximums and Minimums , the direct e ff ects must outweigh the cross e ff ects Rules for two variable Maximums and Minimums 1. Maximum f xx < 0 f yy < 0 f yy f xx f xy f yx > 0 2. Minimum f xx > 0 f yy > 0 f yy f xx f xy f yx > 0 3. Otherwise, we have a Saddle Point From our second order conditions, above, f xx = 2 < 0 f yy = 2 < 0 f xy = 1 f yx = 1 and f yy f xx f xy f yx = ( 2)( 2) (1)(1) = 3 > 0 therefore we have a maximum.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern