= Solution
Because $X$ is a <Noetherian scheme>, choose a finite affine cover $X=\bigcup_iU_i$. Every intersection $U_i\cap U_j$ is quasi-compact, so choose a finite affine cover $U_i\cap U_j=\bigcup_kU_{ijk}$. If $j_i:U_i\hookrightarrow X$ and $j_{ijk}:U_{ijk}\hookrightarrow X$ are the inclusions, the sheaf axiom gives an exact sequence
$$
0\longrightarrow\mathcal F
\longrightarrow\bigoplus_i(j_i)_*(\mathcal F|_{U_i})
\longrightarrow\bigoplus_{i,j,k}(j_{ijk})_*(\mathcal F|_{U_{ijk}}),
$$
where the last arrow is the difference of the two restrictions to each overlap chart.
Applying the left-exact <direct image sheaf> functor identifies $f_*\mathcal F$ with the kernel of
$$
\bigoplus_i(fj_i)_*(\mathcal F|_{U_i})
\longrightarrow
\bigoplus_{i,j,k}(fj_{ijk})_*(\mathcal F|_{U_{ijk}}).
$$
Every map $U_i\to Y$ and $U_{ijk}\to Y$ is a morphism between <affine schemes>. Part (a) shows that all sheaves in the two finite sums are <quasi-coherent sheaves>. Kernels of morphisms of quasi-coherent sheaves on the affine scheme $Y$ are quasi-coherent, because they correspond to kernels of module homomorphisms. Therefore $f_*\mathcal F$ is quasi-coherent. This is the <quasi-coherence of direct image under a quasi-compact quasi-separated morphism> in the present Noetherian case.
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