For any function $f\colon\{0,1\}^n\to\{0,1\}$, define a language $S_f = \{(b_1,b_2,\ldots ,b_n)\in\{0,1\}^n : f(b_1,b_2,\ldots ,b_n) = 1\}$.
So all words in the langugage has same length $n$. I have to show that there exists a boolean function $f$ such that any DFA accepting exactly $S_f$ has at least $\frac{2^{n-2}}{n-2}$ number of states.
My professor said that the fact $x\log(x)$ on $[1,\infty]$ is monotonically increasing would be helpful. Can anyone suggest hints??