infimum and supremum of two argument function in particular set

**waytogo** · November 12th 2010, 09:17 AM

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:
$\frac{\partial f}{\partial x}=2x+2y, <br /> \frac{\partial f}{\partial y}=2x+2y$
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?

**roninpro** · November 12th 2010, 09:18 AM

Have you considered using Lagrange Multipliers?

Pauls Online Notes : Calculus III - Lagrange Multipliers

**waytogo** · November 12th 2010, 09:35 AM

Frankly, I haven't. Now when you mention it, I guess that we have used this method in Economics, unfortunately without proof. Checked the link you added and it seems that it will work for me. Thank you!

But how do you think about my actions before? Unnecessary?

**roninpro** · November 12th 2010, 06:43 PM

I think that it works - it's just painful.

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