Implicit function theorem and Taylor polynomial

Question

Prove that equation $$x\ln w+w\ln y=0$$ sits in a neighborhood of $(x_0,y_0)=(1,1)$, variable $w$ as function of $x$ and $y:$ $w=g(x,y)$.

Prove that $g$ is of class $C^\infty$.

Write second Taylor polynomial of $g$ in $(1,1).$

Answer 1

First, notice that if ${\partial g(z) \over \partial z} = h(z)$, and $h$ is of class $C^r$, then $g$ is of class $C^{r+1}$.

To avoid a notational morass, write $f(w,z) = z_1 \ln w + w \ln z_2$, and note that $f$ is smooth. We have $f(1,(1,1)) = 0$. The implicit function gives some differentiable $g$ defined on a neighbourhood of $(1,1)$ such that $g((1,1)) = 1$ and we have ${\partial g(z) \over \partial z} = - {1 \over {\partial f(g(z),z) \over \partial w}} {\partial f(g(z),z) \over \partial z}$.

Note that $z \mapsto - {1 \over {\partial f(g(z),z) \over \partial w}} {\partial f(g(z),z) \over \partial z}$ is $C^1$ and so $g$ is at least $C^2$, and so $z \mapsto - {1 \over {\partial f(g(z),z) \over \partial w}} {\partial f(g(z),z) \over \partial z}$ is $C^2$. Now repeat...