I need some help with the completing the task given in the last sentence of this old exam question.
"The system
$$
x+y+z&=6 \\
x^2+y^2+z^2&=14
$$
is satisfied at the point $(1,2,3)$. Show that $x$ and $y$ can be solved in a neighborhood of $(1,2,3)$ as a function of $z$. Calculate also $x'(3)$ and $y'(3)$, where $x$ and $y$ are regarded as functions of $z$.".
Any clues?
Let $$f : (0,1) \to \mathbb{R}$$ and $$g(x) = |f(x)|^{r-1} f(x)$$$r \in \mathbb{N}$. It is known that $g\in L^2(0,1)$ and the $r^{th}$ derivative, $ g^{(r)} \in L^2(0,1)$. Show that the first derivative $f^{(1)} \in L^2(0,1)$
Let $f : B_0(r) \setminus \{0\} \to \Bbb C$ be holomorphic, where $B_0(r) \setminus \{0\}$ is the punctured disc of radius $r$.
Assume that $f$ has an essential singularity at $0$.
Is there any way to find $\operatorname{Res}\left(\frac {f'} f, 0\right)$ from the Laurent series of $f$ at $0$?
