Symmetric Matrices & Quadratic Forms · Hard problem

A function of three variables has a critical point with Hessian $H = \begin{pmatrix}-7 & -3 & -3\\-3 & -7 & -3\\-3 & -3 & -7\end{pmatrix}$. Verify that this is a local MAXIMUM by checking that every eigenvalue of $H$ is negative, then find the LARGEST (least negative) eigenvalue.