Let $U$ be some domain in ${\mathbb{R}}^{n}$ with $0\in U$. We have some ${C}^{2}$ function $f:U\to \mathbb{R}$ with $\mathrm{\nabla}f(0)=\mathbf{0}$, therefore 0 is a critical point (i.e. local maximum, local minimum or saddle point). Define $\mathrm{\nabla}f(0)=\mathbf{0}$ and ${f}^{+}:{U}^{+}\to \mathbb{R},x\mapsto f(x)$.

Is $0$ an extremum of ${f}^{+}$ and if yes how could we prove it?

Is $0$ an extremum of ${f}^{+}$ and if yes how could we prove it?