Determine the min, max, and saddle points of F(x) on the unit sphere.

Question

$F(x)=x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}+x_{2}x_{3}+x_{3}^{2}$

$x\in \mathbb{R}^{3}$

unit sphere: $x_1^2+x_2^2+x_3^2=1$

I am trying to find the min, max and saddle points of $F(x)$ on the unit sphere.

I came up with:

$F_{max} = 1+\frac{1}{\sqrt{2}}$ with vectors $\pm (\frac{1}{2},\frac{1}{\sqrt{2}},\frac{1}{2})$

$F_{min} = 1-\frac{1}{\sqrt{2}}$ with vectors $\pm (\frac{1}{2},\frac{-1}{\sqrt{2}},\frac{1}{2})$

I am stuck with how to find the saddle points.

Answer 1

One of the ways is to find eigenvalues for that point. And if the eigenvalues are of opposite signs then it is a saddle point.