Find critical points of $f(x,y,z) = 3x - y -3z$ with constraints $ x^2 + 2z^2 = 225\ ,\ x + y - z = 7$

Question

Find critical points of $f(x,y,z) = 3x - y -3z$ with constraints $ x^2 + 2z^2 = 225\ ,\ x + y - z = 7$

My attempt:

Denote $g_1(x,y,z)=x^2 + 2z^2 -225, g_2(x,y,z)=x + y - z -7$.

$L_1=f+\lambda_1g_1 ,L_2=f+\lambda_2g_2 $.

$\frac{dL_1}{dx}=3+2\lambda_1x=0$

$\frac{dL_1}{dy}=-1=0$

$\frac{dL_1}{dz}=-3+4\lambda_1z=0$

$\frac{dL_1}{d\lambda_1}=x^2+2z^2-225=0$

How am I supposed to deal with $\frac{dL_1}{dy}=-1=0$ ?

$\frac{dL_2}{dx}=3+\lambda_2=0$

$\frac{dL_2}{dy}=-1+\lambda_2=0$

$\frac{dL_2}{dz}=-3-\lambda_2=0$

$\frac{dL_2}{d\lambda_2}=x+y-z-7=0$

In addition I got $\lambda_2=-3 and \lambda_2=1$

I cant get where I am wrong , appreciate any help.

Thanks !

Answer 1

Indeed, $-1=0$ is a problem. It represents the way you can increase or decrease $f$ while keeping the constraint $g_1$ fixed (since $g_1$ doesn't depend on $y$). Put the Lagrangian as $$L=f+\lambda_1g_1+\lambda_2g_2$$ then proceed as you have been.