Let `f(x+1/y) +f(x-1/y) =2f(x) f(1/y) AA x, y in R , y!=0` and f(0)=0 then the value of `f(1) +f(2)=`

Options

A. `-1`
B. 0
C. 1
D. none of these

Answer

B

Explanation

`f(x+(1)/(y))+f(x-(1)/(y))=2f(x)f((1)/(y))AA x, y in R`
Given f(0) = 0
Putting `x=0, y=(1)/(x)`, we get
`f(x)+f(-x)=2f(0)f(x)`
`rArr" "f(x)+f(-x)=0`
`rArr" "f(x)=-f(-x)`
Putting x = 1, y = -1, we get
`rArr" "f(2)+f(0)=2[f(1)]^(2)`
`rArr" "f(2)=2[f(1)]^(2)`
Putting `x=-1, y=-1,` we get
`rArr" "f(-2)=2f(-1)f(-1)`
`rArr" "-f(2)=2(f(1))^(2)`
`rArr" "f(2)=-f(2)`
`rArr" "f(2)=0`
`rArr" "f(1)=0`
`therefore" "f(1)=f(2)=0`
`therefore" "f(1)+f(2)=0`