Question 11
(SWE 2) A function $f:I\to\mathbb R$, defined on an interval $I$, is called concave if $f(\theta x+(1-\theta)y)\ge\theta f(x)+(1-\theta)f(y)$ for all $x,y\in I$ and $0\le\theta\le1$. Assume that the functions $f_1,\ldots,f_n$, having all nonnegative values, are concave. Prove that the function $(f_1f_2\cdots f_n)^{1/n}$ is concave.