 Solution

For $f=x^2+y^2+z^2$ and $g=x^2+2y^2-z^2=1$, the Lagrange multiplier equations areThe four constrained stationary points areOn the constraint, $f=2x^2+3y^2-1$ with $x^2+2y^2\ge 1$. The two $y$-axis points are global minima, while the $x$-axis points are saddles: motion along the waist decreases $f$, whereas motion away from it increases $f$. There is no maximum.

If instead $g=x^2+2y^2-z^2$ is optimized on $x^2+y^2+z^2=1$, the stationary points are all six coordinate-axis points. The $y$-axis points are maxima with value $2$, the $z$-axis points are minima with value $-1$, and the $x$-axis points are saddles with value $1$.