Suppose that $g:[0,1]\to\mathbb{R}$ is a differentiable function with $g(0)=0$. I want to find conditions on $g$ such that $$ \int_0^1 f(x)g(x)\int_x^1f(t)\,dt\,dx=\int_0^1f(t)\int_0^xf(x)g(x)\,dx\,dt>0 $$ for any strictly increasing function $f:[0,1]\to[0,\infty)$ with $f(0)=0$.
Clearly, this holds for $g\geq 0$ and $g(x_0)>0$ for some $x_0\in (0,1)$. Can we find a more general condition?