Positivity of a normal derivative of $C^2$ function

Question

Let $\Omega$ be an open set of $R^n$ of classe $C^2$, let $f \in {C^2}({\bar \Omega } )$ which satisfies the following properties: $f>0$ on $\Omega$ and $|\nabla f|>0$ on $\bar \Omega$. and $f=0$ on $\partial \Omega $. Let $n$ be the outward normal on $\partial \Omega $.

My question: do we have $$\frac{{\partial f}}{{\partial n}} > 0?$$ Thanks.

Answer 1

If $q\in\partial\Omega$, $$\frac{\partial f}{\partial n}(q)=-\lim_{h\to 0^+}\frac{f(q-hn)-f(q)}{h}=-\lim_{h\to 0^+}\frac{f(q-hn)}{h}\leq 0.$$ Also, since $f=0$ on $\partial\Omega$, $$\left|\frac{\partial f}{\partial n}(q)\right|=|\nabla f(q)|>0,$$ hence $\displaystyle\frac{\partial f}{\partial n}(q)<0$.