Find functions $f(x)$ and $g(x)$ such that the following conditions are satisfied for all $x > 0$:

Question

1. $0 < f(x) < 1$,

2. $g(x) < \frac{f(x)}{x} < c$ for some constant $c$,

3. $\frac{d}{dx}g(x) > 0$.

Answer 1

Consider $f(x) = 1/2$. Then we just need $g$ to be a function that approaches $0$ asymptotically from below. The easiest that comes to mind is

$$g(x) =-\frac{1}{x} $$

Answer 2

take $f(x)=\frac{x^2}{x^2+1}$, which is bounded between 0 and 1, and take $g(x)= \arctan(x) - 10$, which is strictly increasing.