The Rayleigh quotient for a symmetric matrix $A$ is $R(x) = \frac{x^T Ax}{x^T x}$ for $x \neq 0$. For $A = \begin{pmatrix} 5 & 1 \\ 1 & 3 \end{pmatrix}$, find the largest eigenvalue (which equals $\max_x R(x)$).