Consider `f: R^+vecRs u c ht h a tf(3)=1` for `a in R^+a n df(x)dotf(y)+f(3/x)f(3/y)=2f(x y)AAx ,y in R^+dot` Then find `f(x)dot`

Answer

`f(x)=1`

Explanation

`f(x)*f(y)+f((3)/(x))f((3)/(y))=2f(xy)`
Put `x=y=1.` Then,
`f^(2)(1)+f^(2)(3)=2f(1)`
or `(f(1)-1)^(2)=0 " or "f(1)=1`
Now, put `y=1`. Then,
`f(x).f(1)+f((3)/(x))f(3)=2f(x)`
or `f(x)=f((3)/(x)) AA x gt 0 " (1)" `
or `f(x)f((3)/(x))+f((3)/(x)) f(x)=2f(3)`
or `f(x)f((3)/(x))=1 " (2)" `
Therefore, from (1) and (2),
`f^(2)(x)=1 AA x gt 0`
` :. f(x)=1AA x gt 0 " " ( :. f(3)=1)`