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Topic: Help with max/min problem
Replies: 5   Last Post: Sep 21, 2003 8:16 PM

 Messages: [ Previous | Next ]
 John E. Prussing Posts: 301 Registered: 12/6/04
Re: Help with max/min problem
Posted: Sep 20, 2003 7:50 PM

In <20b84b19.0309201056.382cb71e@posting.google.com> stegen123@yahoo.com (Jack Smith) writes:

>Hello,

>I need help on this question:

>Find the maximum and minimum values of f(x1,x2) = x1^2 -2x1x2 + 1 over
>D= (x element R^2: ||x||<=1}

>Now I know critical points occur where the gradient is 0. So I
>calculated the gradient and found it is 0 at (0,0). Now I also know I
>need to check the boundary of D for max/min value aswell. I tried
>doing this, but I am stuck. Can some one show me how I can check the
>boundary for max/min values??

>Thanks

A technique that works in general is to use a Lagrange multiplier, b.
Form the function g(x1,x2,b) = f(x1,x2) + b(x1^2 + x2^2 -1). Off the
boundary of D, b=0 and that corresponds to the solution you found.

To determine max/min values on the boundary, determine the critical
points of g. The algebraic sign of the (nonzero) multiplier b determines
whether the point is a constrained max or min.
--
John E. Prussing
University of Illinois at Urbana-Champaign
Department of Aerospace Engineering
http://www.uiuc.edu/~prussing

Date Subject Author
9/20/03 Jack Smith
9/20/03 Virgil
9/20/03 Justin Davis
9/20/03 John E. Prussing
9/21/03 John E. Prussing
9/20/03 Artur