Let f :(0,1) to (0,1) be a differentiabl...

Let `f :(0,1) to (0,1)` be a differentiable functino such that `f '(x) ne 0` for all `x in (0,1) and f ((1)/(2)) =(sqrt3)/(2).` Suppose for all `x lim _(t to x) ((int _(0)^(1) sqrt(1-(f (s))^(2))ds-int _(0)^(x)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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