m-CNF formulas satisfiability approximation
If such a constant-factor approximation algorithm would exist, and it would run in polynomial-time, you could use it to solve CNF-SAT in polynomial time. Since CNF-SAT is NP-hard, this may give you the desired result.

Sketch:
Given a CNF-formula f, feed sufficiently many copies of the formula to your approximation algorithm. If f is not satisfiable, the algorithm must always output 0. If f is satisfiable,  it should output a number > 0 (at least, if the other numbers are chosen correctly).