# Reducing 3SAT to MAX-3SAT

I have the following problem:

Consider the MAX-3-SAT problem: given a Boolean function in Conjunctive Normal Form (CNF) determine the maximum number of clauses that can be satisfied. Prove that this problem is NP-hard.

I know that I'd have to reduce 3-SAT to MAX-3SAT, but I'm pretty lost on how that would work. The related decision problem I figured would be, given clauses and a number k, is there an assignment satisfying at least k of the clauses?

Hint: You have a 3SAT formula $$\varphi$$ in CNF with $$n$$ clauses and a maximal $$m \le n$$ number of clauses can be satisfied. What happens if $$m < n$$? And what about $$m = n$$?