Question 65
For $0 \le x, y \le 1$, let
$$
f(x, y) = x y^2 \sqrt{1-x^2} - x^2 y \sqrt{1-y^2}
$$
Find the minimum constant $c$ for which the following condition holds.
For any integer $n \gt 1$ and any real numbers $a_1, a_2, \dots, a_n$ such that $0 \le a_1 \lt a_2 \lt \dots \lt a_n \le 1$,
$$
f(a_1, a_2) + f(a_2, a_3) + \dots + f(a_{n-1}, a_n) \lt c
$$
$$
f(x, y) = x y^2 \sqrt{1-x^2} - x^2 y \sqrt{1-y^2}
$$
Find the minimum constant $c$ for which the following condition holds.
For any integer $n \gt 1$ and any real numbers $a_1, a_2, \dots, a_n$ such that $0 \le a_1 \lt a_2 \lt \dots \lt a_n \le 1$,
$$
f(a_1, a_2) + f(a_2, a_3) + \dots + f(a_{n-1}, a_n) \lt c
$$