Invertibility of bordered Hessian

Question

I have an optimization problem: $max_{x \in C} f(x)$ s.t. $Ax=b$, where $x \in R^n$ and $b \in R^m$, $m \le n$, adn $C$ compact. I know that $f$ is strictly quasi-concave, and that $A$ has rank $m$ (linearly independent, linear constraints). My question is: can I guarantee that the Hessian matrix bordered with the Jacobian of the constraints is invertible --when all constraints are binding, i.e, the multipliers positive--? That is, can I guarantee that the solution (in $x$ and Lagrange multipliers, which exists and is unique) is differentiable with respect to $b$? Thanks!!!