<s>[INST] Extrema of $x+y+z$ subject to $x^2 - y^2 = 1$ and $2x + z = 1$ using Lagrange Multipliers 
Find the extrema of $x+y+z$ subject to $x^2 - y^2 = 1$ and $2x + z = 1$ using Lagrange multipliers.

So I set it up:
$$
1 = 2x\lambda_1 + 2\lambda_2 \\
1 = -2y\lambda_1 \\
1 = \lambda_2
$$
Plug in for $\lambda_2$:
$$
1 = 2x\lambda_1 + 2 \\
1 = -2y\lambda_1 \\
$$
So we work with:
$$
1 = 2x\lambda_1 + 2 \\
1 = -2y\lambda_1 \\
1 = x^2 - y^2 \\
1 = 2x + z
$$
After some algebra I got $x = y$ as a solution but that's impossible because of the constraint $1 = x^2 - y^2$. What am I missing?
 [/INST] Consider the constraints: you can simplify them in
$$ y=\pm\sqrt{y^2+1}, \qquad z=\mp 2\sqrt{y^2+1}+1$$
for all $y\in\mathbb{R}$.
In this way you can convert the constrained optimization in two unconstrained optimizations: substitute the two solutions in the objective function and you will find 
$$- \sqrt{y^2+1}+y+1, \qquad \sqrt{y^2+1}+y+1.$$
These two real functions are monotone increasing so they don't possess any stationary point.
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