= Solution
{c}
\b[Failure in general.] Let $f:\mathbb A^1_k=\operatorname{Spec}k[t]\to\operatorname{Spec}k$ be the <structure morphism> and take the trivial <line bundle> $\mathcal L=\mathcal O_{\mathbb A^1_k}$. The <polynomial ring> $k[t]$ is a <Noetherian ring>, so this is a <coherent sheaf>. Its <direct image> on the one-point target has sections
$$
\Gamma(\operatorname{Spec}k,f_*\mathcal L)=k[t].
$$
A <coherent sheaf> on $\operatorname{Spec}k$ corresponds to a finite-dimensional <vector space> over $k$, whereas $1,t,t^2,\ldots$ are <linearly independent>. Thus
$$
\boxed{f_*\mathcal O_{\mathbb A^1_k}\text{ is not coherent}.}
$$
\b[Quasi-coherence for the stated morphism.] Now let $f$ satisfy (a), and let $\mathcal F$ be a <quasi-coherent sheaf> on $X$. Over any <affine open subscheme> $V=\operatorname{Spec}A$ of $Y$, write $f^{-1}V=\operatorname{Spec}B$ and $\mathcal F|_{f^{-1}V}=\widetilde M$ for a $B$-<module> $M$. Regard $M$ as an $A$-<module> through the <ring homomorphism> $A\to B$. For $a\in A$,
$$
\Gamma(D(a),f_*\mathcal F)
=\Gamma\bigl(D(\varphi(a)),\widetilde M\bigr)
=M_{\varphi(a)}=M_a.
$$
The last two <localizations> are identical: both invert the action of the same element on $M$. These identifications commute with restrictions, so on the basis of <principal open subsets> they identify $(f_*\mathcal F)|_V$ with the <affine module sheaf> $\widetilde{{}_AM}$. Hence this <direct image> is a <quasi-coherent sheaf> on each $V$, and therefore on $Y$. This part uses only that $f$ is an <affine morphism>.
\b[Coherence under the Noetherian hypotheses.] If $X,Y$ are <Noetherian schemes> and $\mathcal F$ is a <coherent sheaf>, then $M$ is a <finitely generated module> over $B$. Choose generators $m_1,\ldots,m_s$ of $M$ over $B$, and generators $b_1,\ldots,b_r$ of $B$ over $A$, supplied by (a). Then the finite family
$$
\{b_jm_\ell:1\leq j\leq r,\ 1\leq\ell\leq s\}
$$
generates $M$ as an $A$-<module>: expand each coefficient of an expression $m=\sum_\ell c_\ell m_\ell$ in the $b_j$. Since $A$ is a <Noetherian ring>, the <affine module sheaf> $\widetilde{{}_AM}$ is a <coherent sheaf>. This holds on an <affine open cover> of $Y$, proving
$$
\boxed{f_*:\operatorname{QCoh}(X)\to\operatorname{QCoh}(Y),\qquad
f_*:\operatorname{Coh}(X)\to\operatorname{Coh}(Y)\text{ if }X,Y\text{ are Noetherian}.}
$$
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