Let us first recall the logistic cost function
$$f_1(x)=\log(1+e^x)$$
and the negative entropy function
$$f_2(y)=y\log(y)+(1-y)\log(1-y).$$
Here $\log$ denotes the natural logarithm and $x\in \mathbb R$ and $y\in (0,1)$.
We also recall the Fenchel conjugate function $f^*$ of function $f$ as
$$f^*(y):=\sup_{x\in \mathbb R} \, \, yx - f(x).$$
According to Wiki, we know that $$f_1^*(y)=f_2(y).$$
The notion of logistic cost and negative entropy have their own meaning and seem to be independent. I am so surprising when the Fenchel conjugate can establish their relation! Is there an intuition behind such relation?