Constrained Optimization with Equality Constraint problem

a. Consider the problem,

max x1 + x2 + x3 +x4 subject to :

(1)1/3 x1 + 1/3 x2 + 1/8 x3 + 1/8 x4 = 3

(2)x1x2x3x4 = 144

(3)x1, x2, x3, x4 are all greater or equal to 0

i. Write down the necessary condition for the solution of this problem.

ii. Show that the necessary conditions imply x1 = x2 and x3 = x4. Find the solution.

iii. Suppose that the Right Hand Side (RHS) value of the second constraint is changed to x1 x2

x3 x4 = 145. How the change in RHS value of the second constraint will affect the optimal

value of x1 + x2 + x3 + x4, without redoing the whole problem.