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there is no function satisfying the minimum

Source: Iran PPCE 2012-Analysis exam-P3

February 14, 2012
functionintegrationSupportreal analysisreal analysis unsolved

Problem Statement

Consider the set
A={f∈C1([−1,1]):f(−1)=−1,f(1)=1}\mathbb A=\{f\in C^1([-1,1]):f(-1)=-1,f(1)=1\}.
Prove that there is no function in this function space that gives us the minimum of S=∫−11x2f′(x)2dxS=\int_{-1}^1x^2f'(x)^2dx. What is the infimum of SS for the functions of this space?
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