Diverge test, a function powered by itself

Question

question

I don't how the hint can be generalized to common functions f(x).

Answer 1

i think we can use that $$e^x\geq x-1$$ for all real $x$ writing $$f(n)^{f(n)}=e^{\log(f(n)\cdot f(n)}\geq (\log(f(n))-1)\cdot (f(n)-1)$$