RO-NMO 2007 Problem 121

Let $\mathcal{F}$ be the set of functions $f: [0,1]\to\mathbb{R}$ that are differentiable, with continuous derivative, and $f(0)=0$ , $f(1)=1$ . Find the minimum of $\int_{0}^{1}\sqrt{1+x^{2}}\cdot \big(f'(x)\big)^{2}\ dx$ (where $f\in\mathcal{F}$ ) and find all functions $f\in\mathcal{F}$ for which this minimum is attained.