Let $g: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a $C^1$ function, and $x_*$ be a solution of the equation $g(x)=0$.
Suppose $\|J_g(x_*)\| > 0$ where $J_g(x_*)$ is the Jacobian at $x_*$. Show that there exist a positive constant $\epsilon$ such that
$$ \|g(x)\| \leq 6 \|J_g(x_*)\|\|x-x_*\| $$
for all $x$ with $\|x-x_*\| < \epsilon$.
The prove must follow Taylor's expansion but the expansion, as far as I know, is
$$ g(x) = g(x_*) + J_g(x_*)(x-x_*) + o(\|x-x_*\|) $$
Since $g(x_*)$ is zero, we have
$$ g(x)= J_g(x_*)(x-x_*) + o(\|x-x_*\|) $$
How can we come up with coefficient $6$?