I've been struggling with a problem for a while, I have to proove if the following proposition is true or false:
Let $f:\mathbb{R^n}\to\mathbb{R}$ be a smooth funcion (i.e $f \in C¹$). Suppose that $p\in \mathbb{R^n}$ satisfies $f(p) = 0$ and $\nabla f(p) \neq 0 \implies $f has an infinite number of roots.
At first, I thought about $f(x,y)=x^2 + y^2$ at $p=(0,0)$ but it doesn't satisfy $\nabla f(0,0)\neq0$. I also thought about $f(x,y)=\sqrt{x^2 + y^2}$ but it's not smooth.
The only thing that I figured out is that if $\nabla f(p)\neq0 \implies p$ can't be a maximum, minimum or saddle point.
Any hint towards the resolution of this problem would be really appreciated
It's false for $n=1$. For $n > 1$, use the implicit function theorem.