Question

How do you find critical points of multivariable function `f(x,y) =x^3 + xy - y^3`?

Answer 1

Several notations and explanations are available. Here's one:

Find the partial derivatives, set them equal to zero and solve the resulting system of equations.

`f(x,y) =x^3 + xy - y^3`

`f_x = 3x^2 + y = 0`
`f_y = x - 3y^2 = 0`

From the first equation: `y = -3x^2`.

So, we need: `x - 3(-3x^2)^2 = 0`

`x-27x^4 = 0` when `x=0` , in which case `y = -3(0)^2 = 0`

and also when `x=1/3` in which case `y = -1/3`

The critical points are: `(0, 0)` and `(1/3, -1/3)`.

(I've heard that there is an alternative terminology that would find the values of `f` and say that critical points are points in 3-space: `(0,0,0)` and `(1/3, -1/3, -1/27)`)