# Let U be some domain in R^n with 0 in U. We have some C^2 function f:U->R with grad f(0)=0, therefore 0 is a critical point (i.e. local maximum, local minimum or saddle point). Define grad f(0)=0 and f^+:U^+->R,x|->f(x). Is 0 an extremum of f^+ and if yes how could we prove it?

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 $\mathrm{\nabla }f\left(0\right)=\mathbf{0}$, therefore 0 is a critical point (i.e. local maximum, local minimum or saddle point). Define $\mathrm{\nabla }f\left(0\right)=\mathbf{0}$ and ${f}^{+}:{U}^{+}\to \mathbb{R},x↦f\left(x\right)$.
Is $0$ an extremum of ${f}^{+}$ and if yes how could we prove it?
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zastenjkcy
No. As a counterexample, take a function ${f}_{0}:\mathbb{R}\to \mathbb{R}$ with a saddle at the origin (e. g. ${f}_{0}\left(x\right)={x}^{3}$) and then
$f:{\mathbb{R}}^{2}\to \mathbb{R},\phantom{\rule{1em}{0ex}}f\left(x,y\right)={f}_{0}\left(x\right),$
its constant extension in $y$-direction. Then the point $0$ is a critical point of $f$ and its cutoff ${f}^{+}$ to non-negative second coordinate $\left\{y\ge 0\right\}$ still does not have an extremum in $\left(x,y\right)=\left(0,0\right)$ (because $f\left(x,0\right)={f}_{0}\left(x\right)$ has a saddle in $x=0$).