I was reading a book and it says that the sufficient condition for a function to be quasiconcave is that its Bordered Hessian matrix is negative definite. I can't seem to understand this. Please help!
$$ Bordered Hessian= \begin{bmatrix} 0 & f_1 & f_2 \\ f_1 & f_{11} & f_{12} \\ f_2 & f_{21} & f_{22} \end{bmatrix} $$
A submatrix of a negative semidefinite matrix is negative semidefinite. Therefore, the hessian itself is negative semidefinite, meaning the function is concave. Any concave function is quasiconcave.