= Solution
Write $\dot x^\mu=dx^\mu/ds$ and $L=\sqrt{-g_{\mu\nu}\dot x^\mu\dot x^\nu}$. Varying $T=\int L\,ds$ with fixed endpoints and then choosing <proper time> as parameter, for which $L=1$, gives the <Euler-Lagrange equation>
$$
\frac d{d\tau}(g_{\mu\nu}\dot x^\nu)
-\frac12\partial_\mu g_{\alpha\beta}\dot x^\alpha\dot x^\beta=0.
$$
Expanding the derivative, raising the free index with the <inverse metric>, and using the symmetry of $\dot x^\alpha\dot x^\beta$ gives the <geodesic equation>
$$
\boxed{\ddot x^\rho+\Gamma^\rho{}_{\alpha\beta}
\dot x^\alpha\dot x^\beta=0},
\qquad
\boxed{\Gamma^\rho{}_{\alpha\beta}
=\frac12g^{\rho\mu}
(\partial_\alpha g_{\mu\beta}+\partial_\beta g_{\mu\alpha}
-\partial_\mu g_{\alpha\beta})}.
$$
These are the <Christoffel symbols> of the <Levi-Civita connection>.
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