$2-6.$ Consider the following problem

Optimize: $-x_1^2-2x_1+x_2^2$

Subject $to$: $x_1^2+x_2^2-10$

a$.$ Obtain the equation set to be solved to locate the stationary points of the above problem using the method of Lagrange Multipliers. Convert the inequality constraint to an equality constraint with the slack variable $x_3$ as $x_3^2$; why?

b$.$ Show that the following are solutions to the algebraic equations obtained in $($a$).$

$\backslash$table$[[$Stationary Points$],[,$A$,$B$,$C$,$D$],[x_1,-\frac{1}{2},-\frac{1}{2},-1,1],[x_2,\frac{\sqrt[\phantom{2}]{3}}{2},-\frac{\sqrt[\phantom{2}]{3}}{2},0,0],[x_3,0,0,0,0],[,-1,-1,0,2]]$

c$.$ Based on the value of the function being optimized, state whether stationary points A through $D$ are maximum, minimum, or saddle.