lagrange multipliers and the cases involved

Question

I am stuck on the case $x=0$ or $y=0$ or $z=0$

let $f(x,y,z)= x^2+y^2+z^2$ and $g(x,y,z)= x^4+y^4+z^4=11$

Then:

$2x=4x^3\lambda$

$2y=4y^3\lambda$

$2z=4z^3\lambda$

if $x \neq 0, y \neq 0, z\neq 0$ then:

$4x^3yz\lambda=4y^3xz\lambda=4z^3\lambda$ divide by $4\lambda$

$x^3yz=y^3xz=z^3xy$

$x^3yz=y^3xz$ divide by z:

$x^3y=y^3x \to x=z$ Doing the same thing to the other equations gives:

$x=y=z$

let $x=y=z=a$ then $3a^4=11 \to a=\pm\frac{11^{\frac{1}{4}}}{3^{\frac{1}{4}}}$ which gives im not sure what because the algebra is just god awful here.

let x=0

Now what do I do?

Answer 1

$x=y=z=0$ is not a solution.

If $x=y=0$ and $z \ne0$

$z=\pm 11^{1/4}$

If $x=0$ and $y.z\ne 0$ then, because of $2y=4y^3\lambda$ and $2z=4z^3\lambda$ we have

$y=\pm z=\pm(11/2)^{1/4}$

The problem is simmetric for $x,y,z$.