Let f be a non-negative function in [0,1] and twice differentiable in ( 0,1 ). If ∫_0^x √(1-(f^′( t ))^2) dt =∫_0^x f( t ) dt, 0 ≤ x ≤ 1 and f(0)=0,…

Let $\displaystyle f$ be a non-negative function in $\displaystyle [0,1]$ and twice differentiable in $\displaystyle ( 0,1 )$. If $$\int_0^x \sqrt{1-\left(f^{\prime}(\mathrm{t})\right)^2} \mathrm{dt}=\int_0^x f(\mathrm{t}) \mathrm{dt}, 0 \leq x \leq 1 \text { and } f(0)=0 \text {, then } \lim _{x \rightarrow 0} \frac{1}{x^2} \int_0^x f(\mathrm{t}) \mathrm{dt}: $$