which establishes the value

9 as a maximum.

Example 3 Continuing on to Example 2 of Sec. 12.2, we see that Z, = 2xl — A and Z2 = 2x2 - 4A. These yield Zn =2, Z12 = Z21 = 0, and Z22 = 2. From the constraint xx + 4x2 = 2, we obtain g, = 1 and g2 = 4. It follows that the bordered Hessian is

and the value z = A is a minimum

«-Variable Case

When the objective function takes the form z =f(xi, x2,..., x„) subjectto g(x}, x2,..., x„) = c the second-order condition still hinges on the sign of d2z. Since the latter is a constrained quadratic form in the variables dxu dx2,..., dxn, subject to the relation

the conditions for the positive or negative definiteness of d2z again involve a bordered Hessian. But this time these conditions must be expressed in terms of the bordered principal minors of the Hessian. Given a bordered Hessian

0 0

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