Via Lagrange multipliers:
[tex]L(x,y,\lambda)=xy+\lambda(x^2+y^2-4)[/tex]
[tex]L_x=y+2\lambda x=0[/tex]
[tex]L_y=x+2\lambda y=0[/tex]
[tex]L_\lambda=x^2+y^2-4=0[/tex]
[tex]yL_x=y^2+2\lambda xy=0[/tex]
[tex]xL_y=x^2+2\lambda xy=0[/tex]
[tex]\implies yL_x-xL_y=y^2-x^2=0\implies y^2=x^2[/tex]
[tex]\implies x^2+y^2=4=2x^2\implies x^2=2\implies x=\pm\sqrt2\implies y=\pm\sqrt2[/tex]
So we have four critical points to consider, [tex](\sqrt2,\sqrt2),(-\sqrt2,\sqrt2),(\sqrt2,-\sqrt2),(-\sqrt2,-\sqrt2)[/tex]. If both coordinates are positive or both are negative, we get a maximum value of [tex](\pm\sqrt2)^2=2[/tex]; otherwise, we get a minimum of [tex](-\sqrt2)(\sqrt2)=-2[/tex].
