The Lagrangian
![L(x,y,z,\lambda)=x^2+y^2+z^2+\lambda(x^4+y^4+z^4-13)](https://tex.z-dn.net/?f=L%28x%2Cy%2Cz%2C%5Clambda%29%3Dx%5E2%2By%5E2%2Bz%5E2%2B%5Clambda%28x%5E4%2By%5E4%2Bz%5E4-13%29)
has critical points where the first derivatives vanish:
![L_x=2x+4\lambda x^3=2x(1+2\lambda x^2)=0\implies x=0\text{ or }x^2=-\dfrac1{2\lambda}](https://tex.z-dn.net/?f=L_x%3D2x%2B4%5Clambda%20x%5E3%3D2x%281%2B2%5Clambda%20x%5E2%29%3D0%5Cimplies%20x%3D0%5Ctext%7B%20or%20%7Dx%5E2%3D-%5Cdfrac1%7B2%5Clambda%7D)
![L_y=2y+4\lambda y^3=2y(1+2\lambda y^2)=0\implies y=0\text{ or }y^2=-\dfrac1{2\lambda}](https://tex.z-dn.net/?f=L_y%3D2y%2B4%5Clambda%20y%5E3%3D2y%281%2B2%5Clambda%20y%5E2%29%3D0%5Cimplies%20y%3D0%5Ctext%7B%20or%20%7Dy%5E2%3D-%5Cdfrac1%7B2%5Clambda%7D)
![L_z=2z+4\lambda z^3=2z(1+2\lambda z^2)=0\implies z=0\text{ or }z^2=-\dfrac1{2\lambda}](https://tex.z-dn.net/?f=L_z%3D2z%2B4%5Clambda%20z%5E3%3D2z%281%2B2%5Clambda%20z%5E2%29%3D0%5Cimplies%20z%3D0%5Ctext%7B%20or%20%7Dz%5E2%3D-%5Cdfrac1%7B2%5Clambda%7D)
![L_\lambda=x^4+y^4+z^4-13=0](https://tex.z-dn.net/?f=L_%5Clambda%3Dx%5E4%2By%5E4%2Bz%5E4-13%3D0)
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
![xL_x+yL_y=2(x^2+y^2)+52\lambda=-\dfrac2\lambda+52\lambda=0\implies\lambda=\pm\dfrac1{\sqrt{26}}](https://tex.z-dn.net/?f=xL_x%2ByL_y%3D2%28x%5E2%2By%5E2%29%2B52%5Clambda%3D-%5Cdfrac2%5Clambda%2B52%5Clambda%3D0%5Cimplies%5Clambda%3D%5Cpm%5Cdfrac1%7B%5Csqrt%7B26%7D%7D)
![\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
![xL_x+yL_y+zL_z=2(x^2+y^2+z^2)+52\lambda=-\dfrac3\lambda+52\lambda=0\implies\lambda=\pm\dfrac{\sqrt{39}}{26}](https://tex.z-dn.net/?f=xL_x%2ByL_y%2BzL_z%3D2%28x%5E2%2By%5E2%2Bz%5E2%29%2B52%5Clambda%3D-%5Cdfrac3%5Clambda%2B52%5Clambda%3D0%5Cimplies%5Clambda%3D%5Cpm%5Cdfrac%7B%5Csqrt%7B39%7D%7D%7B26%7D)
![\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)