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Math Help - Classifying Critical Points in R3

  1. #1
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    Classifying Critical Points in R3

    f(x,y) = (x^2 + y^2)e^(x^2-y^2) = z
    I have found that the part ∂z/∂y = 0 when y = 0 or +/- sqrt(1 - x^2)
    and that ∂z/∂x = 0 when x=0 or x^2 + y^2 + 1 = 0 which is not real
    so now how do I classify them? i.e. what is min, what is max, what is saddle?
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  2. #2
    MHF Contributor chiph588@'s Avatar
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  3. #3
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    which points am I testing though? I'm not sure how to write them
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  4. #4
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    \displaystyle f_x=2xe^{x^2-y^2}+(x^2+y^2)e^{x^2-y^2}(2x)=2xe^{2x-y^2}(1+x^2+y^2)=0

    \displaystyle x=0, \ 1+x^2+y^2=0

    \displaystyle f_y=2ye^{x^2-y^2}+(x^2+y^2)e^{x^2-y^2}(-2y)=2ye^{x^2-y^2}(1-x^2-y^2)=0

    \displaystyle y=0, \ 1=x^2+y^2

    By substitution,

    \displaystyle x^2+y^2+x^2+y^2=0\Rightarrow 2x^2+2y^2=0\Rightarrow x^2+y^2=0

    Critical point:

    \displaystyle f(0,0)=0\Rightarrow (0,0,0)

    \displaystyle f_{xx}(0,0)=2

    \displaystyle f_{yy}(0,0)=2

    \displaystyle f_{xy}(0,0)=2

    \displaystyle d=f_{xx}f_{yy}-[f_{xy}]^2=4-4=0

    Inconclusive. Those, being f_{xx}, \ f_{yy}, \ f_{xy}, derivatives were pretty nasty though so you might want to double check them.
    Last edited by dwsmith; December 15th 2010 at 10:36 PM. Reason: Clarification, verbage
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  5. #5
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    so (0,0,0) is inconclusive
    but aren't there other possibilities with 1 = x^2 + y^2
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  6. #6
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    No, would (\pm 1,0) or (0,\pm 1) satisfy 1+x^2+y^2=0\mbox{?}
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  7. #7
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    I decided to upload the image since it was interesting when I graphed.
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