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integral of (1+x^2)f'(x)^2

Source: VJIMC 2009 2.2

June 12, 2021
calculusintegration

Problem Statement

Let EE be the set of all continuously differentiable real valued functions ff on [0,1][0,1] such that f(0)=0f(0)=0 and f(1)=1f(1)=1. Define J(f)=∫01(1+x2)f′(x)2dx.J(f)=\int^1_0(1+x^2)f'(x)^2\text dx. a) Show that JJ achieves its minimum value at some element of EE. b) Calculate min⁡f∈EJ(f)\min_{f\in E}J(f).
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