If `f((1)/(x)) +x^(2)f(x) =0, x gt0 and I= int_(1//x)^(x) f(t)dt, (1)/(2) le x le 2x`, then I is equal to

Options

A. `f(2)-f((1)/(2))`
B. `f((1)/(2))-f(2)`
C. 0
D. none of these

Answer

C

Explanation

We have,
`I=underset(1//x)overset(x)intf(t)dt`
`rArr I=underset(x)overset(1//x)intf((1)/(u))(1)/(u^(2))du, "where" t=(1)/(u)`
`rArr I=underset(x)overset(1//x)int-(u^(2)f(u))/(u^(2))du" " [ :. f((1)/(x))=-x^(2)f(x)]`
`rArr I=underset(x)overset(1//x)int f(u) du`
`rArr I=underset(x)overset(1//x)int f(u)du rArr I=- I rArr 2I=0 rArr I=0`