What are the critical points of #f(x,y) =x^3 + xy - y^3#?

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Answer 1 · 2016-04-08T23:50:56

They are #(0,0)# and #(1/3, -1/3)#

For #f(x,y) =x^3 + xy - y^3#, we have

#f_x = 3x^2+y#
#f_y = x-3y^2#.

We need to solve the system

#3x^2+y = 0#
#x-3y^2 = 0#.

The first equation gives us #y = -3x^2#.

Substituting for #y# in the second equation gets us

#x-3(-3x^2)^2 = 0#

#x-27x^4 = 0#

#x(1-27x^3) = 0#

#x=0# #" "# OR #" "# #x=1/3#.

Using #y = -3x^2# from above we get critical points #(0,0)# and #(1/3, -1/3)#.