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Finding local max, min and saddle points

  1. Oct 3, 2011 #1
    1. The problem statement, all variables and given/known data


    2. Relevant equations

    3. The attempt at a solution

    I started out by expanding and got:


    Then I found all my partial derivatives and second derivatives:

    [itex]f_{x}=1+2xy+y^2, f_{y}=1+2xy+x^2, f_{xx}=2y, f_{yy}=2x, f_{xy}=2(x+y)[/itex]

    I know that both first partial derivatives must equal zero so I get:

    [itex]f_{x}=1+2xy+y^2=0[/itex] and [itex]f_{y}=1+2xy+x^2=0[/itex]

    This is the part I am stuck at; I can't find the critical points. I notice that there is symmetry so I tried subtracting the equations but I got y=x and got:


    I also tried setting [itex]f_{x}[/itex] and [itex]f_{y}[/itex] equal to each other but that didn't seem to work.

    Thanks in advance for the help
    Last edited by a moderator: Oct 3, 2011
  2. jcsd
  3. Oct 3, 2011 #2
    I think I may have got it. When I got that [itex]x^2=y^2[/itex] I didn't account for that y could equal (-x). I did that and it got the right answer.
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