$(Second-$Order Conditions $-$ The Bordered Hessian$)*$

Solve the following problem.

Find the local optima $of$ the function $f(x_1,x_2)=x_1-x_2$

subject $to$ the constraint $x_1{x}_2+c=0,$ where $c>0is$ a scalar.

$(a)$ local min: $(x_1,x_2)=(\sqrt[\phantom{2}]{c},-\sqrt[\phantom{2}]{c}),$ local max: $(x_1,x_2)=(-\sqrt[\phantom{2}]{c},\sqrt[\phantom{2}]{c})$

$(b)no$ local min, local max: $(x_1,x_2)=(\sqrt[\phantom{2}]{c},-\sqrt[\phantom{2}]{c})$

$(c)$ local min: $(x_1,x_2)=(-c^{\frac{1}{3}},c^{\frac{2}{3}}),$ local max: $(x_1,x_2)=(c^{\frac{2}{3}},-c^{\frac{1}{3}})$

$(d)$ local min: $(x_1,x_2)=(-c^{\frac{1}{3}},c^{\frac{2}{3}}),no$ local max

$(e)$ local min: $(x_1,x_2)=(-c^{\frac{3}{4}},c^{\frac{1}{4}}),$ local max: $(x_1,x_2)=(c^{\frac{3}{4}},-c^{\frac{1}{4}})$

$(a)$

$(b)$

$(c)$

$(d)$

$(e)$