Help with figuring out if MAX(L) is a CF language
Let $L = a^*b^*c^* \setminus \{ a^n b^n c^k : k &gt; n \}$. To see that $L$ is context-free, note that we can write it as follows:
$$
L = \{ a^n b^m c^k : n &gt; m \} \cup \{ a^n b^m c^k : m &gt; n \} \cup \{ a^n b^m c^k : k \leq n \}.
$$
The only way in which a word in $L$ cannot be extended non-trivially into another word in $L$ is if the word is of the form $a^nb^nc^n$ for $n \geq 1$. Therefore
$$
\mathrm{MAX}(L) = \{ a^n b^n c^n : n \geq 1 \},
$$
which is known to be non-context-free.