# Four stationary points of a multivariable function

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)

#### Solution Preview

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(1) minus (2) to get

3(x^2- y^2)+6(x-y)=0 Thus

(x+y)(x-y)+2(x-y)=0 Thus

(x+y+2)(x-y)=0 Thus

either x=y or y=-x-2

If y=x, then (1) becomes

3x^2+18x+12x=0 This implies

either x=0 or x=10. ...