Is there an example such that $\frac{\partial f}{\partial y}$ vanishes at some point but we can solve for $y$ in terms of $x$ near the point and ...?

Question

I am reading "Analysis on Manifolds" by James R. Munkres.

On p.77,

Example 4. Let $f:\mathbb{R}^2\to\mathbb{R}$ be given by the equation $$f(x,y)=y^2-x^4.$$ Then $(0,0)$ is a solution of the equation $f(x,y)=0$. Because $\frac{\partial f}{\partial y}$ vanishes at $(0,0)$, we do not expect to be able to solve for $y$ in terms of $x$ near $(0,0)$. In fact, however, we can do so, and we can do so in such a way that the resulting function is differentiable. However, the solution is not unique.

I had the following question:

Is there an example such that $f(x,y)=0$ at some point and $\frac{\partial f}{\partial y}$ vanishes at the point but we can solve for $y$ in terms of $x$ near the point and the resulting function is differentiable and unique near the point?

Answer 1

What about $$f(x,y)=y^3-x^9$$ around the origin? Assuming you are only interested in real solutions.

Or $$f(x,y)=(x-y)^2?$$ A double root makes the partial derivative vanish :-)

Answer 2

$f(x,y)=y^3-x$ around the origin would work.