Using the method of Lagrange multipliers, we have the Lagrangian
[tex]L(x,y,z,\lambda_1,\lambda_2)=x+2y+\lambda_1(x+y+z-4)+\lambda_2(y^2+z^2-4)[/tex]
with partial derivatives (set equal to 0)
[tex]L_x=1+\lambda_1=0\implies\lambda_1=-1[/tex]
[tex]L_y=2+\lambda_1+2\lambda_2y=0\implies\lambda_2y=-\dfrac12[/tex]
[tex]L_z=\lambda_1+2\lambda_2z=0\implies\lambda_2z=\dfrac12[/tex]
[tex]L_{\lambda_1}=x+y+z-4=0[/tex]
[tex]L_{\lambda_2}=y^2+z^2-4=0[/tex]
From [tex]L_y[/tex] and [tex]L_z[/tex], we find that [tex]\lambda_2y=-\lambda_2z\implies y=-z[/tex]. Then substituting into [tex]L_{\lambda_2}[/tex], we find
[tex]y^2+z^2=4\implies2y^2=4\implies y=\pm\sqrt2\implies z=\mp\sqrt2[/tex]
and substituting these into [tex]L_{\lambda_1}[/tex], we get
[tex]x+y+z=4\implies x=4[/tex]
So we have two possible critical points, [tex](4,\pm\sqrt2,\mp\sqrt2)[/tex], which give extreme values of [tex]4+2\sqrt2[/tex] and [tex]4-2\sqrt2[/tex], respectively.
