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Math Help - Maximum and minimum problem

  1. #1
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    Maximum and minimum problem

    Hi there. I've got this function f(x,y)=(y-3x^2)(y-x^2), and I have to analyze what happens at (0,0) in terms of maxims and minims. But what I actually have to proof is that theres a saddle roof at that point.

    Theres is a critical point at (0,0). Lets see:

    f(x,y)=(y-3x^2)(y-x^2)=y^2-4yx^2+3x^4

    f_x=-8yx+12x^3
    f_y=2y-4x^2
    Its clear there that there is a critical point at (0,0)

    The determinant of the partial derivatives of second order at (0,0)
    f_{xx}=-8y+36x^2,
    f_{xy}=-8x=f_{yx},
    f_{yy}=2

    f_{xx}(0,0)=0

    \left| \begin{array}{ccc}0 \ 0 \\ 0  \ 2 \\ \end{array} \right| =0

    Then I can't say anything from there. And actually, if I try at any line that passes through (0,0) I would find a minimum. I know that it isn't a minimum, but I don't know how to prove it.

    Bye there.
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  2. #2
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    The second derivative test should do the trick.

    The second derivative test states:

    D=D(a,b)=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^2

    1. If D > 0 and f_{xx}(a,b) > 0, then f(a,b) is a local min.
    2. If D > 0 and f_{xx}(a,b) < 0, then f(a,b) is a local max.
    3. If D < 0, then f(a,b) is a saddle point.
    4. If D = 0, then the test is inconclusive.
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  3. #3
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    The thing is that the determinant gives zero.
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  4. #4
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    The "Taylor's series" for a function of two variables, f(x,y), at (0, 0), is f(0, 0)+ f_x(0, 0)x + f_y(0, 0)y+ \frac{f_{xx}(0, 0)}{2}x^2+  \frac{f_{xy}(0,0)}{2}xy+ \frac{f_{yy}(0, 0)}{2}y^2+ terms of higher degree. Since you have already determined that the first derivatives at (0, 0) are 0, you are saying that the Taylor's series, at (0, 0), is f(0, 0)+ \frac{0}{2}x^2+ \frac{0}{2}xy+ \frac{2}{2}y^2+ terms of higher degree= f(0, 0)+ y^2+ terms of higher degree.

    You can see from that that the " y^2" term has a positive coefficient and so the graph curves upward in the y direction. Since the other second degree terms have 0 coefficient you need to look at the third degree terms which means you need to look at f_{xxx}(0, 0), f_{xyy}(0, 0), etc. If any of them are negative, then you have a saddle point and there is no maximum or minimum at (0, 0). If all of them are positive, then there is a minimum at (0, 0, 0).
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