(a) Let T> 0. A model in economics seeks to optimise investment M(t) by calculating a stationary path of the functional S[M] over the fixed time interval 0 t T where (b) T S[M] = [ dt a(bM M' c), M(0) = Mo, and where a, b, c, Mo are positive constants and M' = dM/dt. (i) Calculate the stationary path M* of S[M]. (ii) Calculate S[M] and M*[T]. (iii) Show that M gives global minimum of the functional S[M]. Let a 0. Consider the functional S[y] = dx xy2, y(a) = 0, subject to the constraint a C[y] =d dx xy = 1. (i) By using the Rayleigh-Ritz method with trial function z = z(x; A, ) = A(a-x), where A > 0 and 3 > 1/2, find an approximation to the smallest eigenvalue of the associated Sturm-Liouville system. (ii) By an appropriate choice of , find the closest approximate value of the eigenvalue from this family of trial functions