Mathematics

How do you find the coordinates of the stationary points to this implicit differentiation equation?

x^2y = y^2 + 2x
2xy + x^2yy' = 2yy' + 2
y'(x^2y-2y) = 2-2xy
y' =

2(1-xy)
-------------
y(x^2-2)

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  1. if z = x^2y-y^2-2x

    ∂z/∂x = 2xy-2
    ∂z/∂y = x^2-2y

    If they are both zero (the stationary point), then

    xy=1, so y=1/x
    x^2-2/x = 0
    x^3-2=0
    x = ∛2
    so, y=1/∛2

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