Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be the function given by \begin{align} f(x, y) := x^3 + xy + y^3 + 1 \end{align} Prove that the level set $f^{-1} \left(\{f(p)\}\right)$ for $p= \left(-\frac{1}{3}, -\frac{1}{3}\right)$ is not a submanifold of $\mathbb{R}^2$ of dimension $1$.
We tried using the Hessian matrix and we found that $p$ is a local maximum of f, but we don't know how to use it, any suggestion?