JEE MATHEMATICS MCQ: INCREASING & DECREASING

If $\int_0^x 2 x f^2(t) d t=\left(\int_0^x 2 f(x-t) d t\right)^2$ for $f(1)=1$ and $f(x)$ is continuous function for $x>0$ and $\left\{a_n\right\}$ is a sequence such that $a_{n+1}=a_n+\sqrt{1+a_n^2}$ for $a=0$; if $f(x)$ is an increasing function, then $\lim _{n \rightarrow \infty} \frac{a_k}{2^{n-1}}=$ (where $\left.k=f\left(n^{\sqrt{2}-1}\right)\right)$ is