Let f:(0,1) in (0,1) be a differenttiabl...

Let `f:(0,1) in (0,1)` be a differenttiable function such that `f(x)ne 0` for all `x in (0,1)` and `f((1)/(2))=(sqrt(3))/(2)`. Suppose for all x,
`underset(x to x)lim(overset(1)underset(0)int sqrt(1(f(s))^(2))dxoverset(x)underset(0)int sqrt(1(f(s))^(2))ds)/(f(t)-f(x))=f(x)`
Then, the value of `f((1)/(4))` belongs to

`{(sqrt7)/(4), (sqrt15)/(4)}`

`{(sqrt7)/(3),(sqrt15)/(3)}`

`{(sqrt7)/(2), (sqrt15)/(2)}`

`{sqrt7, sqrt15}`

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