Let $f:(0,\infty)\to \mathbb{R}$ function such that $f\geq0$ and$$\int_{0}^{\infty}f(x)\mathrm{d}x=1.$$Consider $0<t_1<t_2$. My question is: Is it possible find constants $A,B>0$ with $B<1$ both independent of $t_1$ and $t_2$ such that, for all $t_1,t_2$ as above,
$$\int_{t_1}^{t_2}xf(x)\,\mathrm{d}x\leq A+B(t_2-t_1)?$$
The reason for this question is pure curiosity, I apologize in advance for not having a clear context.