Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.
|
|
|
|
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
|
|
|
|
