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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?

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  • $\begingroup$ an isolated local maximum? i.e. did you check that the hessian is negative definite? $\endgroup$ – Tim kinsella Feb 22 '18 at 14:11
  • $\begingroup$ @Timkinsella yes I checked and the eigenvalues are strictly negative, so it's negative definite. But how can I use it? $\endgroup$ – userr777 Feb 22 '18 at 14:53
  • $\begingroup$ If its an isolated local maximum, then there exists an open neighbourhood $U$ of $p$ such that $f^{-1}(f(p))\cap U = \{p\}$, so $f^{-1}(f(p))$ can't be a 1 dimensional sub manifold. $\endgroup$ – Tim kinsella Feb 22 '18 at 14:57

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