# Thread: Find extreme values of f(x,y):

1. ## Find extreme values of f(x,y):

"3.(4 pts)Find the extreme values of:

$f(x,y) = (x^2 + 3y^2)e^{-x^2-y^2}$

on the disk:

$D = { (x, y) | x^2 + y^2 <= 9 }$"

I used Lagrange multipliers to find the possible minimum values at points (+-3, 0) and possible maximum values at points (0, +-3). However, I have been unsuccessful in finding the critical points located on the inner part of the domain. After looking at a 3d graph of the function, I've determined that there are definitely two critical points in the domains that are local maximums. I've tried finding partial derivatives and setting them equal to 0, thus finding where the tangent planes are level, but I continuously run into the equations:
1 - x^2 - 3y^2 = 0
and
3 - x^2 - 3y^2 = 0
To which there are no solutions. The only other critical point I've found, (0,0), can only be a minimum, and not the maximums I'm looking for.

Can anyone make any more suggestions on things to try, or show me how to find the inner maximums? I'm in calc 3, just learning partial derivatives for the first time.

Thanks, David

2. Notice that to find the critical points you make $\nabla f(x,y)=0$ which gives you the equations:

$2x(1-x^2-3x^2)=0$ and $2y(3-x^2-3y^2)=0$ which in turn are satisfied whenever $x=y=0$ or ( $x=0$ and $y=1$) or ( $x=1$ and $y=0$)

3. Originally Posted by Dwill90
"3.(4 pts)Find the extreme values of:

$f(x,y) = (x^2 + 3y^2)e^{-x^2-y^2}$

on the disk:

$D = { (x, y) | x^2 + y^2 <= 9 }$"

I used Lagrange multipliers to find the possible minimum values at points (+-3, 0) and possible maximum values at points (0, +-3). However, I have been unsuccessful in finding the critical points located on the inner part of the domain. After looking at a 3d graph of the function, I've determined that there are definitely two critical points in the domains that are local maximums. I've tried finding partial derivatives and setting them equal to 0, thus finding where the tangent planes are level, but I continuously run into the equations:
1 - x^2 - 3y^2 = 0
and
3 - x^2 - 3y^2 = 0
To which there are no solutions. The only other critical point I've found, (0,0), can only be a minimum, and not the maximums I'm looking for.

Can anyone make any more suggestions on things to try, or show me how to find the inner maximums? I'm in calc 3, just learning partial derivatives for the first time.

Thanks, David

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CB