Function with vanishing partial derivatives up to order j-1 and exactly one (up to order of variables) nonzero partial derivative of order j

Question

Let $n\geq j\geq 2$ and $f:\mathbb{R} ^n\to \mathbb{R}$ be a smooth function such that $f(0)=0$ whose partial derivatives at 0 up to order $j-1 \in \mathbb{N} $ vanish. Let $$\frac{\partial ^j f}{\partial x_1\dots\partial x_j}(0)$$ and the partial derivatives obtained from commuting the variables be the only nonzero partial derivatives of order $j$. My question is whether there is a neighborhood $V\subset \mathbb{R} ^n$ of $0$ such that $$f(x)\neq 0 \text{ for all }x\in V\cap x\in \mathbb{R} ^n \mid x_1,\dots ,x_j>0 $$ as in the one-dimesional case? If not, is there such a neighborhood $V$, if we make the stronger assumption that $f(x)=0$ whenever there is an $i$ with $x_i=0$? I tried it using the multivariate Taylor formula, but I wasn't able to prove either of the statements so far...