$f\left(x\right)=1+x\left(\sin x\right)\left[\cos x\right],0<x\le \pi /2$ ($[.]$  denotes the greatest integer function)

Is continuous in $\left(0,\pi /2\right)$

Is strictly decreasing in $\left(0,\pi /2\right)$

Is strictly increasing in $\left(0,\pi /2\right)$

Has global maximum value 2