Question 84
Let $n$ be a positive integer number and let $a_1, a_2, \dots, a_n$ be $n$ positive real numbers. Prove that $f : [0, \infty) \to \mathbb{R}$, defined by
$$
f(x) = \frac{a_1+x}{a_2+x} + \frac{a_2+x}{a_3+x} + \dots + \frac{a_{n-1}+x}{a_n+x} + \frac{a_n+x}{a_1+x},
$$
is a decreasing function.
$$
f(x) = \frac{a_1+x}{a_2+x} + \frac{a_2+x}{a_3+x} + \dots + \frac{a_{n-1}+x}{a_n+x} + \frac{a_n+x}{a_1+x},
$$
is a decreasing function.