= Solution
A sheaf $\mathcal F$ of $\mathcal O_X$-modules is <quasi-coherent sheaf>[quasi-coherent] when every affine open $U=\operatorname{Spec}A$ has $\mathcal F|_U\cong\widetilde M$ for some $A$-module $M$.
Cover the target $\mathbb P_k^1$ by $D_+(y_0)$ and $D_+(y_1)$. On the first chart put $s=y_1/y_0$; its inverse image is $D_+(x_0)$ with coordinate $t=x_1/x_0$, and the morphism on rings is
$$
k[s]\longrightarrow k[t],
\qquad s\longmapsto t^2.
$$
As a $k[s]$-module,
$$
k[t]=k[t^2]\oplus t,k[t^2]
$$
is free of rank two. The same calculation on the other standard chart uses $s^{-1}\mapsto t^{-2}$. Hence $f_*\mathcal O_X$ is locally free of rank two on the target.
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