I think I have it now, though I'm sure a simpler solution exists.
Then we have the extrema:
So as N increases, phi-zero decreases and we can use small angle approximations to get
So as N increases, the minimum value of f decreases to -1, so if we can show that when N=2, we are done.
So.. when N=2,
Therefore such that
If anyone can see an easier solution, please let me know.