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Math Help - stationary point on a surface

  1. #1
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    stationary point on a surface

    z= 3x^2 + 7x + 2y^2 - 5y -xy + 13
    Find the stationary point and determine the nature of the point(g).
    My question is did any step im do wrongly?

    STATIONARY POINT
    \frac{\partial z}{\partial x} = 6x + 7 - y

    \frac{\partial z}{\partial y} = 4y - 5 - x
    \frac{\partial z}{\partial x} = 0 ,\frac{\partial z}{\partial y} = 0
    6x + 7 - y = 0
    4y - 5 - x = 0
    Then x and y value is:
    x = \frac{-3}{19}
    y = \frac{23}{19}
    Substitute x and y into original equation(z), then get z=\frac{3263}{361}

    stationary point is at ( \frac{-3}{19}, \frac{23}{19}, \frac{3263}{361})

    Now i want find the nature of the point(g):
    g = (\frac{\partial^2 z}{\partial x^2})(\frac{\partial^2 z}{\partial y^2}) -(\frac{\partial^2 z}{\partial x\partial y} )^2
    = 6(4)-(-1)^2
    =23>0
    Stationary point is a minimum point
    Last edited by wkn0524; May 31st 2010 at 02:07 AM.
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  2. #2
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    Quote Originally Posted by wkn0524 View Post
    z= 3x^2 + 7x + 2y^2 - 5y -xy + 13
    Find the stationary point and determine the nature of the point(g).
    My question is did any step im do wrongly?

    STATIONARY POINT
    \frac{\partial z}{\partial x} = 6x + 7 - y

    \frac{\partial z}{\partial y} = 4y - 5 - x
    \frac{\partial z}{\partial x} = 0 ,\frac{\partial z}{\partial y} = 0
    6x + 7 - y = 0
    4y - 5 - x = 0
    Okay so far!

    Then x and y value is:
    x = \frac{-3}{19}
    y = \frac{23}{19}
    Not even close! Solve those equations again!

    Substitute x and y into original equation(z), then get z=\frac{3263}{361}

    stationary point is at ( \frac{-3}{19}, \frac{23}{19}, \frac{3263}{361})

    Now i want find the nature of the point(g):
    g = (\frac{\partial^2 z}{\partial x^2})(\frac{\partial^2 z}{\partial y^2}) -(\frac{\partial^2 z}{\partial x\partial y} )^2
    = 6(4)-(-1)^2
    =23>0
    Stationary point is a minimum point
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  3. #3
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    Ouch, i knew my mistakes...i did a wrong common factor for x and y.

    x =  -1
    y = 1

    Thanks ya
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