= Solution
On a coordinate patch with $f\ne0$, $g_{xx}=1$, $g_{yy}=f^2$ and the only nonzero <metric tensor> <derivative> is $\partial_xg_{yy}=2ff'$. Thus
$$
\boxed{\Gamma^x{}_{yy}=-ff',\qquad
\Gamma^y{}_{xy}=\Gamma^y{}_{yx}=\frac{f'}f}.
$$
All remaining <Christoffel symbols> vanish. In the supplied <Riemann tensor> convention,
$$
R^x{}_{yxy}=\partial_x(-ff')-\Gamma^y{}_{yx}\Gamma^x{}_{yy}
=-(f'^2+ff'')+f'^2,
\qquad
\boxed{R^x{}_{yxy}=-ff''}.
$$
For example, $f=x$ is flat away from the polar-coordinate origin, while $f=\sin x$ gives positive <Gaussian curvature> $-f''/f=1$.
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