A Non linear program question

(7) Consider the problem: (I) minimize I [ z(.)] = 2 / 2'(t)2 at subject to the conditions x(0) = x(7) =0 and the constraint (II) J[x(.)] = / x(t)2 at = 1. Suppose that x : [0, 7] - R is a C2 function that solves the above problem. Let y : [0, 7] - R be any other C2 function such that y(0) = y (7) = 0. Define a (s ) := ( 1. " (2(1 ) + By( 8 )2 at ) " / 2 and i(s ) : = [ ?()+ sy(.), a (s ) a. Explain why a(0) = 1 and i'(0) = 0. b. Show that (III) " (0 ) = [ " ( ) (tax - (tu(that for some constant A, and find a formula for A in terms of x(t). hint: It may simplify things a little to note that i(s) = (a(s))-2 I[x(.) + sy(.)]. c. Show that if x(.) solves problem (1), (II), then x" (t) + Xx(t) =0 for 0