Is $M=\{(x,y)\in \Bbb{R}^2 : x^2 = y^5\}$ a differentiable submanifold?

Question

Let $M=\{(x,y)\in \Bbb{R}^2 : x^2 = y^5\}$. Determine whether or not $M$ is a differentiable submanifold.

I honestly couldn't get anything out of it. What is the standard approach to this problem?

Thank you.

Answer 1

A very powerful technique for such problems, which I have given on here before, is to use the Inverse Function Theorem to prove that a $k$-dimensional submanifold $M\subset\Bbb R^n$ must be, near each point, locally a graph of a smooth function $f\colon U\to\Bbb R^{n-k}$, where $U$ is an open subset of one of the standard $k$-dimensional coordinate planes in $\Bbb R^n$. [So, in your case, you only have to ask: In the neighborhood of each point of $M$, is it a graph of a smooth function either on the $x$-axis or on the $y$-axis?]