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 November 12th, 2010, 09:17 AM #1 Member   Joined: Jan 2010 Posts: 44 Thanks: 0 infimum, supremum of two argument function in particular set Here is the task: Find supremum and infimum of $f(x,y)=x^2+2xy+y^2$, if $(x,y)\in D, D=\{(x,y)\mid 6x^2+y^2-1=0\}$. I have tried two ways: 1) searching for partial derivatives in order to find local extrema, but here is what happens: $\frac{\partial f}{\partial x}=2x+2y, \frac{\partial f}{\partial y}=2x+2y$ So, it means that there is no local extreme points at all? Or all $(x,y)\mid x=-y$ are extreme points? 2) from D follows that $y=\pm \sqrt(1-6x^2)$. Then I can get function $g(x)=f(x,\pm\sqrt(1-6x^2))$ and look for extreme points of g(x). How do you think? Which way could fit? Or am I wrong in both of them?
 November 12th, 2010, 10:26 AM #2 Member   Joined: Jan 2010 Posts: 44 Thanks: 0 Re: infimum, supremum of two argument function in particular \frac{\partial f}{\partial x}=2x+2y, \frac{\partial f}{\partial y}=2x+2y
 November 12th, 2010, 01:54 PM #3 Global Moderator   Joined: May 2007 Posts: 6,524 Thanks: 587 Re: infimum, supremum of two argument function in particular You need to use Lagrange multipliers. http://en.wikipedia.org/wiki/Lagrange_multipliers

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