Prove that

If $ f:(0,+\infty)\to\mathbb{R}$ be a continuous function, then the following are equivalent (for every $ x\in(0,+\infty)$ ).

(1)$ \frac{f(x_4)-f(x_3)}{{x_4}-x_3}\leq\frac{f(x_2)-f(x_1)}{{x_2}-x_1}\;\;;x_4>x_3>x_2>x_1;$

(2) $ \frac{1}{2}(f(x_2)+f(x_1))\leq\frac{1}{x_2-x_1}\int_{x_1}^{x_2}f(u)du\leq f(\frac{x_2+x_1}{2}).$

I know that if $ f$ has the first condition, it means that $ f$ is a concave function and it implies that $ f$ is a midpoint concave. So we have $ \frac{1}{2}(f(x_2)+f(x_1))\leq f(\frac{x_2+x_1}{2})$ . Moreover I know that the condition (2) along with continuity imply that $ f$ is a convex function and it make the condition (1) be held. Also I know that if $ f$ is a continuous function then according to fundamental theorem of calculus, the integral of $ f$ is well-defined. In addition to these mentioned, I tried to prove the above assertion with Mean value theorem fo integrals. But I could not achieve the aim. Can anyone help me.