Suppose that $f(x)$ and $F(x)$ is the PDF and CDF of a distribution, and we know $f(x)>0$ for all $x\ge0$.
Then function $c(r)$ is defined as the following equation (derived from a bellman equation): $$c(r)-\int_{0}^{Q}c(x)f(Q+r-x){\rm d}x =F(Q+r)-1,\quad \forall\ Q>0,\ 0\leq r\leq Q$$
Can we prove that $c(r)<0$ for all $r$?