Find and classify the critical points

Question

I want to know if my soultion correct or not

$f(x,y)=x^3+y^3-63x-63y+12xy$

$∂f/∂x = 3x^2+12y-63$

$∂f/∂y = 3y^2+12x-63$

let $∂f/∂x=0$ and $∂f/∂y=0$

then

$3x^2+12y-63=0$

$3y^2+12x-63=0$

then

$x=21/4-1/4y$

then

$3(21/4-1/4y)^2-63+12y=0$

$y=-7$ or $y=-15$

from values of y

$x=7$ or $x=9$

the critical points is $(9,-15)$ and $(7,-7)$

Is that correct solution ? if correct there is two saddle points right?

Answer 1

Hint

No, from the first equation, you get $y = \dfrac{-x^2 + 21}{4}$.

Substitute that into the second and you should get $\frac{1}{16} \left(x^4-42 x^2+64 x+105\right) = 0$.

This gives four $x$ values of $x = -7, -1, 3, 5$.

Can you continue with the classification?

Note, not all of the critical points may yield a classification, but you would test them all to find out.