I have this function
$$f(x_1,...,x_6)=x_1x_2x_4x_6+x_1x_3x_5x_6-x_1x_2x_3x_4x_5x_6$$
I want to prove that Hessian of the function $f(x_1,...,x_6)$ is degenerate, Note that I found the Hessian matrix of this function $H_f$ such that
Where determinate of Hessian matrix $D(H_f)=-x_1^4 \left(x_2 x_4-1\right) \left(x_3 x_5-1\right) \left(x_2^3 \left(x_3 x_5-1\right){}^2 \left(5 x_3 x_5+3\right) x_4^3+x_2^2 \left(-7 x_3^3 x_5^3+7 x_3^2 x_5^2+3 x_3 x_5-3\right) x_4^2-x_2 x_3 x_5 \left(x_3^2 x_5^2-3 x_3 x_5+6\right) x_4+3 x_3^2 x_5^2 \left(x_3 x_5-1\right)\right) x_6^4$
and $$C=\{(0,x_2,x_3,x_4,x_5,0),(0,0,0,x_4,x_5,x_6),(x_1,0,0,x_4,x_5,0),(0,x_2,0,0,x_5,x_6),(x_1,x_2,0,0,x_5,0), \{x_1,0,0,0,0,x_6)\}$$ represent the set all critical point for f.
since $D(H_F)(c)=0,\forall c$ , where c is a critical point
I used this test to prove that all critical point is saddle point
How can I prove a mathematical proof that that Hessian of the function $f(x_1,...,x_6)$ is degenerate .
How can I prove that all critical point are saddle where $D(H_F)(c)=0,\forall c$ ?
Thanks for the help