[Math] lagrange multipliers fails

Question

I am looking for a certain counter example. Assume a $C^1$ function $f$ is to be optimized with respect to a $C^1$ constraint $g=0$, and we have at a point $(x,y)$, the existence of a lagrange multiplier $\lambda$ with
$$
\begin{align}
&\nabla f(x,y)=\lambda\nabla g(x,y)\\
&\nabla g(x,y)\neq0\\
\end{align}
$$
But $f$ fails to have an extremum at this point with respect to $g=0$

thanks

Answer 1

Let $f(x,y):=x^3+y$ and $g(x,y):=y$. Then $\nabla f(0,0)=\nabla g(0,0)=(0,1)$, but $f(x,0)-f(0,0)=x^3$ assumes both signs in the immediate neighborhood of $(0,0)$.