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

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

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Answer 1 · 2015-04-11T15:49:40

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 $(1/3, -1/3, -1/27)$ )

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

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

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