<s>[INST] Minimizing $x^2+y^2+z^2$ subject to $xy -z + 1 = 0$ via Lagrange multipliers $$\begin{array}{ll} \text{minimize} & f(x,y,z) := x^2 + y^2 + z^2\\ \text{subject to} & g(x,y,z) := xy - z + 1 = 0\end{array}$$

I tried the Lagrange multipliers method and the system resulted from has no solution. So I posted it to see if the question is wrong by itself or I'm missing something.
So I made the Lagrangian equation $L(x,y,z,λ)=x^2 + y^2 + z^2 + λ(xy -z+1)$
and then
$θL/θx = 2x + λy =0$
$θL/θy = 2y + λx =0$
$θL/θz = 2z - λ =0$
$θL/θλ = xy -z +1 =0 $
The obvious solution for that system is x=0 , y=0 , z=1 and λ=2
But solving it in an online solver for nonlinear systems of equation the answer I get is that it's unsolvable.
So my question is: What I'm doing wrong
 [/INST] This can be solved in at least two methods.
First, let's solve without Lagrange, using convenient changes of variables.  Let $u=x+y, v=xy$.  This results in $u^2=x^2+y^2+2xy=x^2+y^2+2v$.
We now need to minimize $u^2-2v+z^2$ under the constraint $v-z+1=0$.  We can rearrange this constraint to be of the form $z=1+v$ and therefore $z^2=1+2v+v^2$.  Substituting this, we need to minimize $u^2+v^2$.  The minimum of this is for $u=0, v=0$, which returns $x=0, y=0, z=1$.
Solving this using Lagrange:
$$L=x^2+y^2+z^2-\lambda(xy-z+1)=x^2+y^2+z^2-\lambda xy-\lambda z-\lambda$$
$$\frac{\partial L}{\partial z}=2z-\lambda\rightarrow\lambda=2z$$
$$\frac{\partial L}{\partial x}=2x-\lambda y=0\rightarrow x=\frac{\lambda y}{2}=yz$$
$$\frac{\partial L}{\partial y}=2y-\lambda x=0\rightarrow2y-2yz^2=0$$$$\rightarrow y=0, x=0\cup z=1,\lambda=2,x=y\cup z=-1,\lambda=-2,x=-y$$
We have three possible solutions to this.  We will plug each into the equation for the constraint $xy-z+1=0$
If $x=0, y=0$, our constraint becomes $-z+1=0$, which has the solution $x=0, y=0, z=1$, with the value of $x^2+y^2+z^2=1$
If $z=1, x=y$, our constraint becomes $x^2-1+1=0$, which has the exact same solution
If $z=-1, x=-y$, cour constraint becomes $-y^2+1+1=0$, which has the solutions $x=\pm\sqrt{2}, y=\mp\sqrt{2}, z=-1$.  The value here is $x^2+y^2+z^2=5$, which is not the minimum
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