6. Let \( f:(0,1) \rightarrow(0,1) \) be a differentiable function such that \( f^{\prime}(x) \neq 0 \) for all \( x \in(0,1) \) and \( f\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2} \). Suppose for all \( x, \lim _{t \rightarrow x}\left(\frac{\int_{0}^{t} \sqrt{1-(f(s))^{2}} d s-\int_{0}^{x} \sqrt{1-(f(s))^{2}} d s}{f(t)-f(x)}\right)=f(x) \). Then the value of \( f\left(\frac{1}{4}\right) \) belongs to : (a) \( \left\{\frac{\sqrt{7}}{4}, \frac{\sqrt{15}}{4}\right\} \) (b) \( \left\{\frac{\sqrt{7}}{3}, \frac{\sqrt{15}}{3}\right\} \) (c) \( \left\{\frac{\sqrt{7}}{2}, \frac{\sqrt{15}}{2}\right\} \) (d) \( \{\sqrt{7}, \sqrt{15}\} \)
