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Math Help - critical points multivariable

  1. #1
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    critical points multivariable

    show that the function f(x,y)=xy^2-x^2y has a unique critical point. what kind is it?

    I tried using the second derivative test. So from first derivatives I got the critical points (0,0) (1,0) and (1,2)

    I then tried using the determinant where D(a,b)=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^2 but it was always inconclusive. For example D=0 for the point (0,0) and f_{xx}=0 for (1,0) and I don't know how to interpret that.

    I haven't learned Larange multipliers yet.
    Last edited by superdude; November 16th 2009 at 10:34 PM. Reason: latex
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  2. #2
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    Quote Originally Posted by superdude View Post
    show that the function f(x,y)=xy^2-x^2y has a unique critical point. what kind is it?

    I tried using the second derivative test. So from first derivatives I got the critical points (0,0) (1,0) and (1,2)

    How? f_x=y^2-2xy=y(y-2x)=0\,,\,\,f_y=2xy-x^2=x(2y-x)=0\,\Longrightarrow\,x=y=0 \Longrightarrow\,(0,0) is the only solution to this system, as can be checked...or better: check that (1,0)\,,\,(1,2) are not zeroes of this sytem but (,0) is.[tex]


    I then tried using the determinant where D(a,b)=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^2 but it was always inconclusive. For example D=0 for the point (0,0) and f_{xx}=0 for (1,0) and I don't know how to interpret that.

    And you don't need to: critical points are points where the first order partial derivatives vanish, and that's what's been done above


    I haven't learned Lagrange multipliers yet.
    You don't need them and anyway you can't use them here.

    Tonio
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  3. #3
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    ok I follow you so far. But that doesn't answer the question what type of critical point is at (0,0)
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  4. #4
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    Quote Originally Posted by superdude View Post
    ok I follow you so far. But that doesn't answer the question what type of critical point is at (0,0)

    No, it doesn't...and according to you that wasn't asked. I suspect it is a saddle point, but you'll have to go into higher partial derivatives (perhaps the Wronskian matrix...I just don't remember) to find out.

    Tonio
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  5. #5
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    could someone define "unique critical point"?
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