Question

The number of solutions for $[\left[x\right]-x]=\sin x$ is (where $[.]$ denotes the greatest integer function) is

• A. $5$
• B. $7$
• C. $0$
• D. infinite

Answer 1

The correct option is D infinite

$f\left(x\right)=[\left[x\right]-x]=[-(x-[x\left]\right)]=[-\left\{x\right\}]$

$\left\{x\right\}=0\forall x\in Z$
$0<\left\{x\right\}<1\in \mathrm{Z/}$

$f\left(x\right)=[-\left\{x\right\}]=\{\begin{array}{cc}0,& x\in Z\\ -1,& x\in \mathrm{Z/}\end{array}$

$\sin x\in [-1,1]$


From graph, it is clear that there are infinite number of solutions.