Isn't the issue that avN = Integrate[x*f[x_]/(1 - F[x]), {x, a0N, 1}] doesn't converge on {a0N, 1} ? (Also, you need to remove the underscore in f[x_].)
avN = Integrate[x*f[x_]/(1 - F[x]), {x, a0N, 1}]
{a0N, 1}
f[x_]
Thanks for the response Jim! I did remove the underscore, thanks for pointing out. It is true that I have this convergence problem but usually Mathematica gives some result anyway. Now it just does not give any output at all. I also tried NSolve which does not work either.
