RO-NMO 2016 Problem 101

Let be a natural number $n\ge 2$ and $n$ positive real numbers $a_1,a_2,\ldots ,a_n$ whose product is $1.$ Prove that the function $f:\mathbb{R}_{>0}\longrightarrow\mathbb{R} , f(x)=\prod_{i=1}^n \left( 1+a_i^x \right) ,$ is nondecreasing.