The Lagrangian

has critical points where the first derivatives vanish:




We can't have
, since that contradicts the last condition.
(0 critical points)
If two of them are zero, then the remaining variable has two possible values of
. For example, if
, then
.
(6 critical points; 2 for each non-zero variable)
If only one of them is zero, then the squares of the remaining variables are equal and we would find
(taking the negative root because
must be non-negative), and we can immediately find the critical points from there. For example, if
, then
. If both
are non-zero, then
, and

![\implies x^2=\sqrt{\dfrac{13}2}\implies x=\pm\sqrt[4]{\dfrac{13}2}](https://tex.z-dn.net/?f=%5Cimplies%20x%5E2%3D%5Csqrt%7B%5Cdfrac%7B13%7D2%7D%5Cimplies%20x%3D%5Cpm%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D)
and for either choice of
, we can independently choose from
.
(12 critical points; 3 ways of picking one variable to be zero, and 4 choices of sign for the remaining two variables)
If none of the variables are zero, then
. We have

![\implies x^2=\sqrt{\dfrac{13}3}\implies x=\pm\sqrt[4]{\dfrac{13}3}](https://tex.z-dn.net/?f=%5Cimplies%20x%5E2%3D%5Csqrt%7B%5Cdfrac%7B13%7D3%7D%5Cimplies%20x%3D%5Cpm%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D)
and similary
have the same solutions whose signs can be picked independently of one another.
(8 critical points)
Now evaluate
at each critical point; you should end up with a maximum value of
and a minimum value of
(both occurring at various critical points).
Here's a comprehensive list of all the critical points we found:
![(\sqrt[4]{13},0,0)](https://tex.z-dn.net/?f=%28%5Csqrt%5B4%5D%7B13%7D%2C0%2C0%29)
![(-\sqrt[4]{13},0,0)](https://tex.z-dn.net/?f=%28-%5Csqrt%5B4%5D%7B13%7D%2C0%2C0%29)
![(0,\sqrt[4]{13},0)](https://tex.z-dn.net/?f=%280%2C%5Csqrt%5B4%5D%7B13%7D%2C0%29)
![(0,-\sqrt[4]{13},0)](https://tex.z-dn.net/?f=%280%2C-%5Csqrt%5B4%5D%7B13%7D%2C0%29)
![(0,0,\sqrt[4]{13})](https://tex.z-dn.net/?f=%280%2C0%2C%5Csqrt%5B4%5D%7B13%7D%29)
![(0,0,-\sqrt[4]{13})](https://tex.z-dn.net/?f=%280%2C0%2C-%5Csqrt%5B4%5D%7B13%7D%29)
![\left(\sqrt[4]{\dfrac{13}2},\sqrt[4]{\dfrac{13}2},0\right)](https://tex.z-dn.net/?f=%5Cleft%28%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%2C%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%2C0%5Cright%29)
![\left(\sqrt[4]{\dfrac{13}2},-\sqrt[4]{\dfrac{13}2},0\right)](https://tex.z-dn.net/?f=%5Cleft%28%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%2C-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%2C0%5Cright%29)
![\left(-\sqrt[4]{\dfrac{13}2},\sqrt[4]{\dfrac{13}2},0\right)](https://tex.z-dn.net/?f=%5Cleft%28-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%2C%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%2C0%5Cright%29)
![\left(-\sqrt[4]{\dfrac{13}2},-\sqrt[4]{\dfrac{13}2},0\right)](https://tex.z-dn.net/?f=%5Cleft%28-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%2C-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%2C0%5Cright%29)
![\left(\sqrt[4]{\dfrac{13}2},0,\sqrt[4]{\dfrac{13}2}\right)](https://tex.z-dn.net/?f=%5Cleft%28%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%2C0%2C%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%5Cright%29)
![\left(\sqrt[4]{\dfrac{13}2},0,-\sqrt[4]{\dfrac{13}2}\right)](https://tex.z-dn.net/?f=%5Cleft%28%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%2C0%2C-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%5Cright%29)
![\left(-\sqrt[4]{\dfrac{13}2},0,\sqrt[4]{\dfrac{13}2}\right)](https://tex.z-dn.net/?f=%5Cleft%28-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%2C0%2C%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%5Cright%29)
![\left(-\sqrt[4]{\dfrac{13}2},0,-\sqrt[4]{\dfrac{13}2}\right)](https://tex.z-dn.net/?f=%5Cleft%28-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%2C0%2C-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%5Cright%29)
![\left(0,\sqrt[4]{\dfrac{13}2},\sqrt[4]{\dfrac{13}2}\right)](https://tex.z-dn.net/?f=%5Cleft%280%2C%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%2C%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%5Cright%29)
![\left(0,\sqrt[4]{\dfrac{13}2},-\sqrt[4]{\dfrac{13}2}\right)](https://tex.z-dn.net/?f=%5Cleft%280%2C%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%2C-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%5Cright%29)
![\left(0,-\sqrt[4]{\dfrac{13}2},\sqrt[4]{\dfrac{13}2}\right)](https://tex.z-dn.net/?f=%5Cleft%280%2C-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%2C%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%5Cright%29)
![\left(0,-\sqrt[4]{\dfrac{13}2},-\sqrt[4]{\dfrac{13}2}\right)](https://tex.z-dn.net/?f=%5Cleft%280%2C-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%2C-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D2%7D%5Cright%29)
![\left(\sqrt[4]{\dfrac{13}3},\sqrt[4]{\dfrac{13}3},\sqrt[4]{\dfrac{13}3}\right)](https://tex.z-dn.net/?f=%5Cleft%28%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%2C%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%2C%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%5Cright%29)
![\left(\sqrt[4]{\dfrac{13}3},\sqrt[4]{\dfrac{13}3},-\sqrt[4]{\dfrac{13}3}\right)](https://tex.z-dn.net/?f=%5Cleft%28%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%2C%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%2C-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%5Cright%29)
![\left(\sqrt[4]{\dfrac{13}3},-\sqrt[4]{\dfrac{13}3},\sqrt[4]{\dfrac{13}3}\right)](https://tex.z-dn.net/?f=%5Cleft%28%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%2C-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%2C%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%5Cright%29)
![\left(-\sqrt[4]{\dfrac{13}3},\sqrt[4]{\dfrac{13}3},\sqrt[4]{\dfrac{13}3}\right)](https://tex.z-dn.net/?f=%5Cleft%28-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%2C%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%2C%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%5Cright%29)
![\left(\sqrt[4]{\dfrac{13}3},-\sqrt[4]{\dfrac{13}3},-\sqrt[4]{\dfrac{13}3}\right)](https://tex.z-dn.net/?f=%5Cleft%28%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%2C-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%2C-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%5Cright%29)
![\left(-\sqrt[4]{\dfrac{13}3},\sqrt[4]{\dfrac{13}3},-\sqrt[4]{\dfrac{13}3}\right)](https://tex.z-dn.net/?f=%5Cleft%28-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%2C%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%2C-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%5Cright%29)
![\left(-\sqrt[4]{\dfrac{13}3},-\sqrt[4]{\dfrac{13}3},\sqrt[4]{\dfrac{13}3}\right)](https://tex.z-dn.net/?f=%5Cleft%28-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%2C-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%2C%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%5Cright%29)
![\left(-\sqrt[4]{\dfrac{13}3},-\sqrt[4]{\dfrac{13}3},-\sqrt[4]{\dfrac{13}3}\right)](https://tex.z-dn.net/?f=%5Cleft%28-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%2C-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%2C-%5Csqrt%5B4%5D%7B%5Cdfrac%7B13%7D3%7D%5Cright%29)