Consider the following set in $\mathbb R^3$: $$ M=\{(x,y_1,y_2):x^3y_1+x^2y_1y_2+x+y_1^2y_2=0\}. $$ a) Show that there are open neighborhoods $U$ of $1\in\mathbb R$ and $V$ of $(-1,1)\in\mathbb R^2$, and a $C^1$-function $\phi\colon V\to U$, such that $$ M\cap(U\times V)=\{(\phi(y_1,y_2),y_1,y_2):(y_1,y_2)\in V\}. $$
b) Determine $D_1\phi(-1,1),D_2\phi(-1,1),D_1D_2\phi(-1,1)$.
c) Show that $M\cap(U\times V)$ is a 2-manifold.
I have a couple of questions, and I'm mostly stuck at b.
a) I am guessing I should invoke the implicit function theorem here. Let $f(x,y_1,y_2)=x^3y_1+x^2y_1y_2+x+y_1^2y_2$. It holds that $$ \dfrac{\partial f(1,-1,1)}{\partial x}\neq 0, $$ and $f$ is $C^1$. So there exists neighboorhoods $U\ni (-1,1)$ and $V\ni 1$ with a unique $C^1$ function $\phi\colon V\to U$, such that $f(\phi(y_1,y_2),y_1,y_2)=0$, and $\phi(-1,1)=1$.
b) Here I don't know what to do, because I don't have $\phi$ explicitly? I tried solving $f(x,y_1,y_2)=0$ for $x$, which would then yield $\phi(y_1,y_2)$, but I had no success.
c) I know that if $Df(x,y_1,y_2)$ has rank 1 on each point of $U$, then $M$ is a manifold, but how am I supposed to work with $U$? Should I just note that $Df$ is continuous, and since it has rank 1 on $(1,-1,1)$, we can find a neighbourhood about $(1,-1,1)$ on which is still has rank 1, and then we just take the intersection with $U$?