Consider the minimization problem with the following condition:

(5) Consider the problem: (1) minimize I[a:(-)] = % Kim)? dt subject to the conditions 32(0) : $(7r) : 0 and the constraint (11) J[:c(-)] = / $(t)2 dt = 1. 0 Suppose that a: : [0, 7r] > R is a 02 function that solves the above problem. Let y : [0,7T] > R be any other 02 function such that 31(0) 2 WT) = 0- Dene and 3. Explain Why 04(0) : 1 and i'(0) : 0. b. Show that (111) z"(0) = f: :c'(t) y'(t) alt A f: a:(t) y(t) dt for some constant A, and nd a formula for A in terms of $(t) hint: It may simplify things a little to note that i(s) : (05(3))i2 I[3:(-) + sy(-)]. c. Show that if J:(-) solves problem (I), (11), then 37"(15) + A376) : 0 for 0