Consider the following convex programming problem:

Maximize f(x) 24x1  x1 2 10x2  x2 2

, subject to x1 8, x2 7, and x1  0, x2  0.

(a) Use the KKT conditions for this problem to derive an optimal solution.

(b) Decompose this problem into two separate constrained optimization problems involving just x1 and just x2, respectively.

For each of these two problems, plot the objective function over the feasible region in order to demonstrate that the value of x1 or x2 derived in part

(a) is indeed optimal. Then prove that this value is optimal by using just the first and second derivatives of the objective function and the constraints for the respective problems.