`[2x]-2[x]=lambda` where [.] represents greatest integer function and {.} represents fractional part of a real number then

Options

A. (a) `lambda=1 AA x in R`
B. (b) `lambda=0 AA x in R`
C. (c) `lambda =1 AA {x} ge (1)/(2) `
D. (d) `lambda =0 AA {x} lt (1)/(2)`

Answer

C, D

Explanation

`[2x]=[x]+[x+(1)/(2)]`
` :. [2x]-2[x]=[x+(1)/(2)]-[x]`
`=[[x]+{x}+(1)/(2)]-[x]`
` =[{x}+(1)/(2)]`
If ` {x} ge (1)/(2) implies {x} +(1)/(2)ge 1implies [{x} +(1)/(2)]=1 implieslambda =1`
If `{x} lt (1)/(2) implies {x} +(1)/(2)lt 1implies [{x} +(1)/(2)]=0 implieslambda =0`