I am guessing that it must be, but I feel that I have only found a unique critical point, so that this point could be either a unique absolute max or a unique absolute min -- I say uniqueness because the question states further to show that the minimum of f is unique.
Do I just conclude that this single critical point is exactly my unique absolute min, without having other critical points to evaluate and compare to? My work included solving a block system of linear equations.
EDIT: The function is: $\large I(u)=∑_{i,j=1}^nL_{ij}u_iu_j−2∑_{j=1}^nf_ju_j$
with the assumption that $∑_{i,j=1}^n L_{ij}u_iu_j≥0$ and strictly positive for all non-zero $\vec u$.
The constraint is: $\large ∑_{j=1}^n A_{ij}u_j = g_i$ for i=1,2,…,m.
The Hessian that I computed in the unconstrained optimization part of the question was positive-definite for all critical points, hence all critical points found were minimums of f. But I do not think that the Hessian is relevant to the constrained optimization part of the problem, for which I have found one critical point.
Thanks,