Show this $|\sin{x}|+|\sin{(x+1)}|+|\sin{(x+2)}|>\frac85$

Question

Let $x\in R$ show that $$f(x)=|\sin{x}|+|\sin{(x+1)}|+|\sin{(x+2)}|>\dfrac{8}{5}$$

since $$f(x)=f(x+\pi),$$it sufficient to show $x\in (0,\pi]$

Answer 1

Since $\sin$ is a concave function on $[0,\pi]$ and sum of concave functions is a concave function,

we have $$\min_{[0,\pi]}f=\min\{f(0),f(\pi-1),f(\pi-2),f(\pi)\}=f(\pi-1)=2\sin1>\frac{8}{5}$$