2. On X = {u Lip([0, 1]): u(0) = 0, u(1) = 1} and for 20 consider the functional F: X → R Fe(u) = √√√¹ + (x + c)u²(x)²dx. [0,1] i) Prove that infx Fo= 1 and deduce that Fo does not have minimum on X. ii) Prove that there exists have minimum on X. >0 such that for all 0 << o the functional F, does not Hints: i) √1+t≤1+√t for t≥0. ii) Euler-Lagrange equation with u(0) = 0, prove that u(1) < 1 for small € > 0.

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