I am trying to find the extreme values of the function $f(x,y,z)=x^2+y^2+z^2$ given the constraints $g(x,y,z)=x-y=1$ and $h(x,y,z)=y^2-z^2=1$.
I am rather lost on this question as it feels as though I am taking the right steps but keep coming to impossible solutions.
Here is my current work:
$$\nabla f = \begin{bmatrix}2x\\ 2y\\ 2z\end{bmatrix}$$ $$\nabla g = \begin{bmatrix}1\\ -1\\ 0\end{bmatrix}$$ $$\nabla h = \begin{bmatrix}0\\ 2y\\ -2z\end{bmatrix}$$ Let $2z = -2bz\implies b = -1$, and $a=2x$; we have $$2y = -a + 2by$$$$2y = -a - 2y$$$$4y = -a$$ $$2x = -4y$$$$x = -2y$$ $x - y = 1\implies$ $$-3y = 1 \implies \begin{cases} y = -\frac{1}{3} \\ x = \frac23\end{cases} $$ Substituting this into $$y^2 - z^2 = 1$$ Gives us $$z^2 = -\frac89$$
A squared value cannot be negative (complex numbers don't make sense in this case). I don't understand where I have gone wrong. I would like to know where I might've gone wrong.