J: C' ([0, 1], IR) → R by J(y) = | e (x) dx, for all y E C' ([0, 1], R). Define Let A = {y € C" (0, 1] | y(0) = 0 and y(1) = 1}, %3| and put V, = C ([0, 1], R). (a) Show that J is Gâteaux differentiable at every y E C'([0, 1], R) in every direction 7 E Vo, and compute dJ(y; ŋ). (b) Find a critical point of J in the class A. (c) Use the inequality e > 1+ z, for all z e R, to deduce that the critical point of J found in part (b) is a minimizer of J over the class A.

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