Question

What are the critical points for g(x,y)=x^3+y^3+(9/(xy)) and are they maximums or minimums? I understand I find the partial derivatives which should be 3x^2-(9/(yx^2)) when fx and 3y^2-(9/(xy^2)) when fy. Then I set those to be equal to zero in order to find the critical points but when I do that I am getting no solution despite knowing there is supposed to be one critical point.

Answer 1

since you want both partials to be zero, you need
3x^2 - 9/(x^2y) = 0
3y^2 - 9/(xy^2) = 0

3x^4y = 9
3xy^4 = 9
so a critical point is at x = y = 3^(1/5)
or at (1.24573,1.24573)