Let $f$ be a differentiable function on $\mathbb{R}$ such that $f(2)=1,\ f'(2)=4$. Let $\displaystyle \lim_{x\to 0}\big(f(2+x)\big)^{\frac{3}{x}}=e^{\alpha}$. Then the number of times the curve $y=4x^3-4x^2-4(\alpha-7)x-\alpha$ meets the $x$-axis is: