For a real number r we denote by [r] the largest integer less than or equal to r. If x,y are real numbers with `x,y ge 1` then which of the following statements is always true? A) `[x+y] le [x]+[y]` B) `[xy] le [x][y]` C) `[2^x] le 2^[x]` D)`[(x)/(y)] le [x]/[y]`

Options

A. `[x+y]le[x]+[y]`
B. `[xy]le[x]+[y]`
C. `[2^(x)]le2^([x])`
D. `[(x)/(y)]le([x])/([y])`

Answer

D

Explanation

(A) `[x+y]le[x]+[y]`
let x=0.1
y=0.9
`[0.1+0.9]le[0.1]+[0.9]`
`1 le 9+0` wrong
(B) `[xy]le[x]+[y]`
`x=2,y=(1)/(2)`
`[2*(1)/(2)]le[2][(1)/(2)]`
`rArr1le0` wrong
(C) `[2^(x)]le2^([x])`
`x=0.99[2^(0.99)]le2^([0.99])`
`[2^(0.99)]le2^(@)=1` wrong
(D) `[(x)/(y)]le([x])/([y])`
given `x,yge1`
if `xlty[(x]/(y)]=0" " 0le([x])/([y])` true
if `xgey[(x]/(y)]le([x])/([y])` always true