Let $f :\mathbb{R}^n \to \mathbb{R}$ be twice continuously differentiable and $\nabla f(x) = 0$ and $H_f(x)$ positive definite. Is it true that $\nabla f$ does not have any other roots in an environment around $x$?
I seem to be missing something trivial here. I believe it's true. By the conditions, $x$ is a strict local minimum/maximum. I'd like to suppose that there would be a sequence of strict minima/maxima converging to $x$ otherwise but where is the contradiction?
The function is strictly convex in a convex neighbourhood of $x$, because the Hessian is positive definite there. The gradient of a strictly convex function is $0$ only at a global minimu, which is unique.