If I can solve Max Clique, can I solve 3-SAT?
Yes.
Construct the graph with k clauses. Ask Max Clique solver for the size of the largest clique. If size == k, then satisfiable.
Security Example: A museum has hallways (edges) and intersections (nodes). What is the minimum number of guards to see every hallway?
Minimum Vertex Cover.
This is NP-Hard.