TR-MO 1994 Problem 4

Let $f: \mathbb{R}^{+}\rightarrow \mathbb{R}+$ be an increasing function. For each $u\in\mathbb{R}^{+}$ , we denote $g(u)=\inf\{ f(t)+u/t \mid t>0\}$ . Prove that: $(a)$ If $x\leq g(xy)$ , then $x\leq 2f(2y)$ ; $(b)$ If $x\leq f(y)$ , then $x\leq 2g(xy)$ .