I was trying to find the maximum, the minimum and the saddle points of the following function:
$f(x,y,z)=x^3 + y^3 + z^3 -9xy -9xz +27x$
I derivate the function, so:
$f_x=3x^2-9y-9z+27$
$f_y=3y^2-9x$
$f_z=3z^2-9x$
So in order to find these point I need to check when $f_x=0$;$f_y=0$ and $f_z=0$; but the only thing I've been able to prove is that y=z by suming the last two but when I tried to use it in the first one I get : $y=\frac{x^2}{6}+\frac{3}{2}$ which does not have any root. I've tried other ways but I always ended up in the same result. And honestly I don't know what to anymore. I would really appreciate any advice you can give me