[tex]L(x,y,z,\lambda_1,\lambda_2)=y+4z+\lambda_1(2x+z-4)+\lambda_2(x^2+y^2-1)[/tex]
[tex]L_x=2\lambda_1+2\lambda_2 x=0\implies\lambda_1+\lambda_2x=0[/tex]
[tex]L_y=1+2\lambda_2y=0[/tex]
[tex]L_z=4+\lambda_1=0\implies\lambda_1=-4[/tex]
[tex]L_{\lambda_1}=2x+z-4=0[/tex]
[tex]L_{\lambda_2}=x^2+y^2-1=0[/tex]
[tex]\lambda_1=-4\implies \lambda_2x=4\implies\lambda_2=\dfrac4x[/tex]
[tex]1+2\lambda_2y=0\implies\lambda_2y=-\dfrac12\implies8y=-x[/tex]
[tex]x^2+y^2=1\implies (-8y)^2+y^2=65y^2=1\implies y=\pm\dfrac1{\sqrt{65}}[/tex]
[tex]y=\pm\dfrac1{\sqrt{65}}\implies x=\mp\dfrac8{\sqrt{65}}[/tex]
[tex]2x+z=4\implies z=4\pm\dfrac{16}{\sqrt{65}}[/tex]
We have two critical points to consider: [tex]\left(-\dfrac8{\sqrt{65}},\dfrac1{\sqrt{65}},4+\dfrac{16}{\sqrt{65}}\right)[/tex] and [tex]\left(\dfrac8{\sqrt{65}},-\dfrac1{\sqrt{65}},4-\dfrac{16}{\sqrt{65}}\right)[/tex].
At these points, we respectively have a maximum of [tex]16+\sqrt{65}[/tex] and a minimum of [tex]16-\sqrt{65}[/tex].
