Let $P(x)$ be a polynomial (of an odd degree $n$) strictly increasing on $(-\infty, +\infty).$ Then $F(x)=\displaystyle \frac{1}{1+\exp\{-P(x)\}}$ is a distribution function of a polynomial logistic distribution associated with $P,$ whose density equals $$ f(x)=\frac{P^\prime(x)\exp\{-P(x)\}}{(1+\exp\{-P(x)\})^2}. $$ This distribution has been introduced in [1].
Clearly, $f(x)\geq 0$ and $\displaystyle \lim_{x\rightarrow\pm\infty}f(x)=0,$ whence $f$ has at least one maximum on $(-\infty, +\infty).$
Question. Given $n,$ what is the greatest number of the points of local maxima for $f$?
Remark. For $n=1,$ there is exactly one local maximum, while for $n=3$ there can be either 1 or 2 local maxima (proved in [2]). I think that this number, $M(n),$ satisfies $1\leq M(n)\leq (n+1)/2.$
REFERENCES: [1] V. M. Koutras, K. Drakos, and M. V. Koutras, A polynomial logistic distribution and its application in finance, Communications in Statistics: Theory and Methods, 43 (10-12), 2045-2065 (2014). Special Issue: Advances in Probability and Statistics.
[2] U. Aksoy, S. Ostrovska, A. Y. Ozban, Polynomial logistic distribution associated with a cubic polynomial, Communications in Statistics: Theory and Methods, in press.
