Question 144
Let $n \in \mathbb{N}$ and $\mathcal{A}$ be a nonempty family of nonempty subsets of $\{1, 2, \dots, n\}$ with the following property – if $A \in \mathcal{A}$ and $A \subset B \subseteq \{1, 2, \dots, n\}$, then $B \in \mathcal{A}$. Prove that the function
$$
f(x) := \sum_{A \in \mathcal{A}} x^{|A|} (1-x)^{n-|A|}
$$
is strictly increasing in the interval $(0, 1)$.
$$
f(x) := \sum_{A \in \mathcal{A}} x^{|A|} (1-x)^{n-|A|}
$$
is strictly increasing in the interval $(0, 1)$.