= Solution
Set $p=t-\lambda r$ using part (a). At every node,
$$
f(y_j)-p(y_j)=\lambda(-1)^j,\qquad \|f-p\|_{L^\infty(Y)}=|\lambda|.
$$
If $\lambda=0$, the error vanishes. Otherwise any degree-at-most-$n$ <polynomial> $q$ with strictly smaller error would make $q-p$ have alternating nonzero signs at the increasingly ordered nodes (reversed globally if $\lambda<0$). It would then have at least $n+1$ distinct roots by the <intermediate value theorem>, impossible. Hence
$$
\boxed{p=t-\lambda r\text{ minimizes }\|f-q\|_{L^\infty(Y)}\text{ over }\deg q\leq n.}
$$
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