4. Let f(x, z) = C ([0, 1] R) be strictly convex with respect to its...

 

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4. Let f(x, z) = C ([0, 1] R) be strictly convex with respect to its second argument and define for y(x) = C[0,1], J (y) = [* f(x, y'(x)) dx. (a) (7 points) Consider J(y; v) on the set D = {y C [0, 1], y(0) = 0}. Is J convex on D? Strictly convex? (b) (2 points) Consider now J on C1 [0, 1]. Is it convex on C1 [0, 1]? Strictly convex? (c) (2 points) Show that a function yo(x) satisfying f(x, y(x)) = 0 on [0,1] minimizes J(y) on C1 [0, 1]. (d) (4 points) Find a function y that minimizes 1 J[y] = [ " ey'(x) exy' (x)dx C over C1 [0, 1] subject to y(0) = 0. Is it the unique minimum of J? 4. Let f(x, z) = C ([0, 1] R) be strictly convex with respect to its second argument and define for y(x) = C[0,1], J (y) = [* f(x, y'(x)) dx. (a) (7 points) Consider J(y; v) on the set D = {y C [0, 1], y(0) = 0}. Is J convex on D? Strictly convex? (b) (2 points) Consider now J on C1 [0, 1]. Is it convex on C1 [0, 1]? Strictly convex? (c) (2 points) Show that a function yo(x) satisfying f(x, y(x)) = 0 on [0,1] minimizes J(y) on C1 [0, 1]. (d) (4 points) Find a function y that minimizes 1 J[y] = [ " ey'(x) exy' (x)dx C over C1 [0, 1] subject to y(0) = 0. Is it the unique minimum of J?