So I am asked to find out which level sets of the function $f:\mathbb{R}^2 \to \mathbb{R}$ given by $f(x,y)=x^3+xy+y^3+1$ are embedded submanifolds of $\mathbb{R}^2$.
You can see that the points in the set $X=\mathbb{R}-\{1,\frac{28}{27}\}$ are regular values ($(0,0)$ and $(-\frac{1}{3},-\frac{1}{3})$ are not regular points).
So $f^{-1}(q)$ is a embedded submanifold for every $q$ in $X$.
My question is, how can I find out if $f^{-1}(1)$ is/is not an embedded submanifold? Likewise with $\frac{28}{27}$.
Thanks in advanced.