Just stated reduces to that pre- nomic Ficient condition is table is adequate. le 12.1 with those der condition in- choice variables grangian function sented in Table 12.1. EXERCISE 12.3 1. Use the bordered Hessian to determine whether the stationary value of z obtained in each part of Exercise 12.2-1 is a maximum or a minimum. 2. In stating the second-order sufficient conditions for constrained maximum and mini- mum, we specified the algebraic signs of | H21, H3, H41, etc., but not of | H₁1. Write out an appropriate expression for | H₁, and verify that it invariably takes the negative sign. 3. Recalling Property II of determinants (Sec. 5.3), show that: (a) By appropriately interchanging two rows and/or two columns of H2 and duly altering the sign of the determinant after each interchange, it can be transformed into Z11 Z12 91 Z21 Z22 92 91 92 0 (b) By a similar procedure, |H3| can be transformed into Z11 Z12 Z13 91 Z21 Z22 Z23 92 Z31 Z32 Z33 93 g 92 93 0 and the double- ivatives of the into four areas What alternative way of "bordering" the principal minors of the Hessian do these results suggest? 4. Write out the bordered Hessian for a constrained optimization problem with four choice variables and two constraints. Then state specifically the second-order sufficient condition for a maximum and for a minimum of z, respectively.