A local maximum for a Quadratic Form over Quadric surface is an absolute maximum?

Question

Let $Q$$_A$ and $Q$$_B$ be quadratic forms defined on R$^n$. Where the associated $n$ x $n$ matrices, $A$ $=$ $A$$^t$ and $B$ $=$ $B$$^t$ are nondegenerate. Consider $Q$$_A$ defined on the quadric surface $S$={$\vec{x}$ $\in$ R$^n$ | $Q$$_B$($\vec{x}$) $=$ $c$} and let's require that $c$ is not $0$ to ensure that $\nabla$$Q$$_B$ $\neq$ $\vec{0}$ for all $\vec{x}$ $\in$ $S$. My question is, if $\vec{a}$ is a point where $Q$$_A$ takes on a local maximum relative to other values on the surface, is $Q$$_A$($\vec{a}$) necessarily an absolute maximum?