Is the following language context free?
$L$ is not context free. You can use the Ogden's lemma to show it.

I'll use the Wikipedia notation linked above.

For every $p$ take the word $a^pb^pc^p$. For a the marking where only c are marked whatever the decomposition $uxyzv$ you take, $ux^iyz^iv$ will not be in $L$.
Four cases: $z\in c^*$ then it fails for $i=0$ or $z\notin c^*$ then it fails for $i&gt;0$. And the symmetric cases for $x$.

Hope it helps.