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    Four stationary points of a multivariable function

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    A surface is described by the multivariable function f(x,y) where:

    f(x,y) = x^3 + y^3 + 9(x^2 + y^2) + 12xy

    a) Show that the four stationary points of this function are located at:

    (x1, y1) = (0, 0)
    (x2, y2) = (-10, -10)
    (x3, y3) = (-4, 2)
    (x4, y4) = (2, -4)

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    Solution Preview

    Stationary points are the points such that the partial derivatives of f(x,y) both with respect to x and y are 0.

    Take the partial derivative of f(x,y) both respect to x and y to get

    3x^2+18x+12y=0 (1)
    3y^2+18y+12x=0 ...

    Solution Summary

    This shows how to find the four stationary points of a function.