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Math Help - Find extreme values of f(x,y):

  1. #1
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    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
    Last edited by CaptainBlack; November 13th 2009 at 10:34 PM. Reason: To change the font to something generic that everyone can read
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  2. #2
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    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)
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Dwill90 View Post
    "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

    When you copy and paste something from a source into the post window please try to remove the font information that gets copied as well. Your original posting specified a font that made you post unreadable to at least some of the users of MHF.

    CB
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