[tex]L(x,y,\lambda)=4x+8y+\lambda(x^2+y^2-5)[/tex]
[tex]L_x=4+2\lambda x=0[/tex]
[tex]L_y=8+2\lambda y=0[/tex]
[tex]L_\lambda=x^2+y^2-5=0[/tex]
[tex]L_y-2L_x=2\lambda y-4\lambda x=2\lambda(y-2x)=0\implies y=2x[/tex]
[tex]xL_x+yL_y=2\lambda(x^2+y^2)+4x+8y=10\lambda+4x+8y=0\iff5\lambda+2x+4y=0[/tex]
[tex]y=2x\implies 5\lambda+5y=0\implies 5(\lambda+y)=0\implies y=-\lambda[/tex]
[tex]4+2\lambda x=0\iff4+\lambda y=0\iff4-y^2=0\implies y=\pm2[/tex]
[tex]y=\pm2\implies x=\dfrac y2=\pm1[/tex]
In other words, there are two critical points, [tex](1,2)[/tex] and [tex](-1,-2)[/tex]. At the first point, there is a maximum of [tex]f(1,2)=20[/tex] and at the second, there is a minimum of [tex]f(-1,-2)=-20[/tex].
