Problem #PRU-73808

Problem

Author: V.A. Popov

On the interval $[0;1]$ a function $f$ is given. This function is non-negative at all points, $f(1)=1$ and, finally, for any two non-negative numbers $x_1$ and $x_2$ whose sum does not exceed 1, the quantity $f(x_1+x_2)$ does not exceed the sum of $f(x_1)$ and $f(x_2)$.

a) Prove that for any number $x$ on the interval $[0;1]$, the inequality $f(x_2)\le 2x$ holds.

b) Prove that for any number $x$ on the interval $[0;1]$, the $f(x_2)\le 1.9x$ must be true?