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!
$$
Bordered Hessian=
\begin{bmatrix}
0 & f_1 & f_2 \\
f_1 & f_{11} & f_{12} \\
f_2 & f_{21} & f_{22}
\end{bmatrix}
$$
Best Answer
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.