$(1)$ Consider the following nonlinear programming problem:

Maximize $f(x_1,x_2)=x_1^2-4x_1+x_2^2-6x_2$

subject $to{x}_1+x_{2}3$

$-2x_1+x_{2}2$

$x_1,x_{2}0$

$($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?