= Solution
The <metric tensor> and its inverse are diagonal: $g_{xx}=1$, $g_{yy}=f^2$, $g^{xx}=1$, $g^{yy}=f^{-2}$ on a patch where $f\ne0$. Only $\partial_xg_{yy}=2ff'$ is nonzero. The <Christoffel symbol> formula therefore gives
$$
\boxed{\Gamma^x_{yy}=-ff',\qquad\Gamma^y_{xy}=\Gamma^y_{yx}=\frac{f'}f,}
$$
with all other components zero. Using the stated <Riemann curvature tensor> convention,
$$
R^x{}_{yxy}=\partial_x\Gamma^x_{yy}-\Gamma^y_{yx}\Gamma^x_{yy}
=-(f'^2+ff'')+f'^2,
$$
so $\boxed{R^x{}_{yxy}=-ff''}$. This is the <curvature of a two-dimensional warped metric>; its <Gaussian curvature> is $-f''/f$.
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