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integral inequality

Source: romanian nmo 2007, grade 12, problem 1

April 15, 2007
calculusintegrationinequalitiesfunctionderivativelogarithmstrigonometry

Problem Statement

Let F\mathcal{F} be the set of functions f:[0,1]→Rf: [0,1]\to\mathbb{R} that are differentiable, with continuous derivative, and f(0)=0f(0)=0, f(1)=1f(1)=1. Find the minimum of ∫011+x2⋅(f′(x))2 dx\int_{0}^{1}\sqrt{1+x^{2}}\cdot \big(f'(x)\big)^{2}\ dx (where f∈Ff\in\mathcal{F}) and find all functions f∈Ff\in\mathcal{F} for which this minimum is attained. [hide="Comment"] In the contest, this was the b) point of the problem. The a) point was simply ``Prove the Cauchy inequality in integral form''.
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