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 $(1/3, -1/3)$ .

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