Minimum number of roots of second derivative

Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x)=x^2-x, x\in\mathbb{R}$. Let $g:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function such that $g(x)=0$ has exactly three distinct roots in the open interval $(0, 1)$. Let $h(x)=f(x)g(x), x\in\mathbb{R}$, and $h"$ be the second order derivative of the function $h$. If $n$ is the number of roots of $h"(x)=0$ in $(0, 1)$, then the minimum possible value of $n$ equals __________