The suff stat is the max, because it cannot be separated from
that's the factorization theorem in use.
NOW, for your answer.
The marginal densities are ZERO if the X's are not in
HENCE, the joint density is ZERO if all the X's are not in
And, note that
is the same as
is the same as and
YOUR teacher should have had indicator functions on the previous lines as well, let me show you