Let `f:(-oo,oo)vec[0,oo)` be a continuous function such that `f(x+y)=f(x)+f(y)+f(x)f(y),AAx in Rdot` Also `f'(0)=1.` Then `[f(2)]` equal `([dot]` represents the greatest integer function`)` `5` b. `6` c. `7` d. `8`

Options

A. 5
B. 6
C. 7
D. 8

Answer

B

Explanation

`f(x+y)=f(x)+f(y)+f(x)f(y), AA x, y in R`
`therefore" "1+f(x+y)=(1+f(x))(1+f(y))`
Put `g(x)=1+f(x)`, we get
`therefore" "g(x+y)=g(x)g(y)`
`therefore" "g(x)=e^(kx)`
`therefore" "f(x)=e^(kx)-1`
`" "f'(0)=1" gives "k=1`
`therefore" "f(x)=e^(x)-1`
`therefore" "[f(2)]=[e^(2)-1]=6`