I'm currently learning how to solve constrained maximisation problems. More specifically, since I have to follow the steps provided by my professor, I have to find out if the solution to a maximisation problem - if any - is unique, before working on the Kuhn-Tucker conditions.
Let $\mathcal{f}:\mathbb{R^3}\rightarrow \mathbb{R}$, $\mathcal{g}:\mathbb{R^3}\rightarrow \mathbb{R}$ be the objective function and the constraint function respectively. Let X $\subseteq$ $\mathbb{R^3}$ be $\mathcal{f}$'s and $\mathcal{g}$'s domain.
Here is what he's written (I re-arranged some things to make it shorter for you to read):
We know that the solution to the problem, if any, is unique if:
$\langle\mathcal{f}$ is strictly quasi-concave $\land$ $\mathcal{g}$ is quasi-concave$\rangle$ $\bigvee$ $\langle\mathcal{g}$ is strictly quasi-concave $\land$ $\mathcal{f}$ is quasi-concave and locally non-satiated $\vee$ $\mathcal{f}$ is affine and non constant $\vee$ $\mathcal{f}$ is quasi-concave and strictly monotone $\vee$ $\mathcal{f}$ is quasi-concave and $\forall$ x $\in$ X, D$\mathcal{f}$(x) $\gg$ 0 $\vee$ $\mathcal{f}$ is quasi-concave and $\forall$ x $\in$ X, D$\mathcal{f}$(x) $\ll$ 0$\rangle$
My main question is: how can I check if e.g. $\mathcal{f}$ is strictly quasi-concave and $\mathcal{g}$ is quasi-concave, using the Hessian matrix?
My preliminary answer: I'd try to check if e.g. $\mathcal{f}$ is strictly concave ($\implies$ strictly quasi-concave) and, if it was so, I'd proceed to check if $\mathcal{g}$ is pseudo-concave ($\implies$ quasi-concave), or vice-versa. The same goes for the other "part" of the quoted conditions, should the one I've just used as an example not be verified. What am I missing? Is there a more direct method to check strict-quasi-concavity and quasi-concavity using the Hessian?
What I know:
strict concavity $\implies$ concavity $\implies$ pseudo concavity $\implies$ quasi concavity.
strict concavity $\implies$ differentiable-strict-quasi-concavity.
Lastly, I know how to use the Hessian to check if a function is strictly concave, concave or quasi-concave.