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