Symmetric Matrices & Quadratic Forms · Superchallenge problem

A real symmetric $3\times 3$ matrix $A$ has eigenvalues $1$, $4$ and $9$. Find the maximum of $(\mathbf x^{T}A\mathbf x)(\mathbf x^{T}A^{-1}\mathbf x)$ over all unit vectors $\mathbf x$.