Question 60
Consider a function $f : \mathbb{R} \to [0, \infty)$. Prove that $f$ satisfies the inequality $f(x+y) \ge (1+y)f(x)$ for any $x \in \mathbb{R}$ and any $y \ge 0$, if and only if the function $g : \mathbb{R} \to [0, \infty)$ defined by $g(x) = e^{-x}f(x)$, for $x \in \mathbb{R}$, is non-decreasing.