Let f be a non–negative function in [0, 1] and twice differentiable in (0, 1).
If \(\int\limits_0^x \sqrt{1-(f'(t))^2}dt = \int\limits_0^xf(t)dt,\) 0 \(\leq\) x \(\leq\) 1 and f(0) = 0, then \(\lim\limits_{x \to \infty}\)\(\cfrac{1}{2}\)\(\int\limits_0^x f(x)dt:\)
(1) equals 0
(2) equals 1
(3) does not exist
(4) equals \(\cfrac{1}{2}\)
