Q:

Lagrange Multipliers to find the maximum and minimum values

Using Lagrange Multipliers I need to find the maximum and minimum values of the function $f(x,y,z)=x^2+y^2+z^2$ subject
to the given constraints: $g(x)= x^2/\alpha+y^2/\beta+z^2/\gamma=1$ and also $\alpha>\beta>\gamma>0$
I come to the conclusion that $x=\pm\alpha$ using the derivatives in the picture and also applying $\alpha^2=\lambda$,              but I'm stuck from this point, can anyone help me continue and find the min max vales?

A:

We can re-write the first three equations as $$2x(1-\frac\lambda{\alpha^2})=0\\2y(1-\frac\lambda{\beta^2})=0\\2z(1-\frac\lambda{\gamma^2})=0$$
Notice that since $\alpha\ne\beta\ne\gamma$ at least two of the $x,y,z$ must be $0$. From the equation of the constraint you see that $x=y=z=0$ is not a solution. Therefore the only possible solutions are

$x=y=0$, $z=\pm\gamma$
$x=z=0$, $y=\pm\beta$
$y=z=0$, $x=\pm\alpha$

The value of $f(x,y,z)$ in these points is $\gamma^2$, $\beta^2$, and $\alpha^2$ for the above cases. The maximum is the last option, the minimum is the first.

