Is the following function concave? Can anyone hellp, please?

Question

Is the function strictly concave $ν(x_1,x_2) = 2x_1 +x_2 − 9/2 ax_1^2 -2ax_2^2 − 6aρx_1x_2 $ over the set $D:= S =\{x_1+x_2=1\}$? Where $a>0$ and $|ρ|\leq1$. Applying the Hessian test, we get that the matrix is NSD(negative semi definite) when $ρ=1$ and ND otherwise. The set $S$ is convex but it is open. Can we apply the Hessian test also to closed sets??

Answer 1

Edited to respect the changes made to the original question.

First note that the domain of interest $D$ is convex.

Taking the Hessian is one way to begin. We have $$ H = \begin{pmatrix} -9a & -6a\rho \\ -6a\rho & -4a \end{pmatrix}\,. $$ Then the second derivative of $\nu$ along the set $D$ is $$ (1,\,-1)\, H \, (1,\,-1)^{\mathrm{T}} = (12\rho -13)\,a < 0\,, \quad \text{since } |\rho|\leq1\,. $$ Hence (by the second derivative condition) we see the function $\nu$ is strictly concave over $D$.