If `f(x)+2f((1)/(x))=3x,x ne 0, and S={x in R: f(x)=f(-x)},` then S

Options

A. is an empty set
B. contains exactly one element
C. Contains exactly two elements .
D. contains more than two elements

Answer

C

Explanation

We have,
`f(x)+2f((1)/(x))=3x" " `….(i)
Replacing x by `(1)/(x)`, we get
`f((1)/(x))+2f(x)=(3)/(x)" " `….(ii)
Solving (i) and (ii) , we obtain
`f(x)=3((2)/(x)-x)`
`:. f(x)=f(-x)`
`implies 3((2)/(x)-x)=3(-(2)/(x)+x)`
`implies (4)/(x)-2x=0implies 2x^(2)-4=0implies x=pm sqrt(2)`
Hence , `S={x in R: f(x)=f(-x)}={-sqrt(2), sqrt(2)}`