Consider the following nonlinear programming problem:

Maximize f$($x$1,$ x$2)=$ x

$2$

$14$x$1+$ x

$2$

$26$x$2$

subject to x$1+$ x$2<=3$

$2$x$1+$ x$2<=2$

x$1,$ x$2>=0$

$($i$)$ Formulate the problem using Lagrange multipliers.

$($ii$)$ What can you say about the Second$-$order optimality conditions?

$($iii$)$ Solve the problem using an appropriate method.

$($iv$)$ Compare the Lagrangian duality formulation with the Wolfe$-$type duality

$($v$)$ Does the Weak Duality Apply?