Consider the problem of Example 4.1.3 in Section 4.1. Calculate the primal function corresponding to the minimum and verify that its gradient is related to the Lagrange multiplier as specified by the sensitivity theorem. Example 4.1.3 (4.22) Consider the problem minimize (a + x3 + x3) subject to x1 + x2 + x3 = 3. The first order necessary conditions (4.20) yield ri+1=0, *+ = 0, 33+ * = 0, xi + x2 + xy = 3. This is a system of four equations and four unknowns. It has the unique solution Xi = x3 = x3 = 1, X = -1. The constraint gradient here is (1,1,1), so all feasible vectors are regular. Therefore, r' = (1,1,1) is the unique candidate for a local minimum. Fur- thermore, since VzL(x", 1") is the identity matrix for this problem, the second order necessary condition (4.21) is satisfied. We can argue that x* is a global minimum by using the convexity of the cost function and the convexity of the constraint set to verify that t satisfies the sufficiency condition of Prop. 3.1.1(b) (alternatively, we can use a sufficiency condition to be given in Section 4.3.2). If instead we consider the maximization version of problem (4.22), i.e., minimize - ] (ai + x3 + x3) (4.23) subject to 11 + 12 + 13 = 3, then the first order condition (4.20) yields x = (1, 1, 1) and X* = 1. However, the second order condition (4.21) is not satisfied and since every feasible vector is also regular, we conclude that the problem has no solution