For a polynomial $g(x)$ with real coefficients, let $m_g$ denote the number of distinct real roots of $g(x)$. Suppose $S$ is the set of polynomials with real coefficients defined by $S = \{(x^{2} - 1)^{2}(a_0 + a_1 x + a_2 x^{2} + a_3 x^{3}) : a_0, a_1, a_2, a_3 \in \mathbb{R}\}$. For a polynomial $f$, let $f'$ and $f''$ denote its first and second order derivatives, respectively. Then the minimum possible value of $(m_{f'} + m_{f''})$, where $f \in S$, is ______

$f(x) = (x^2 - 1)^2\,p(x),\quad p(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3$

$f(\pm 1) = 0 \text{ and } f'(\pm 1) = 0 \ \Rightarrow\ \pm 1 \text{ are double roots of } f$

$\text{To find: } \min_{f\in S}\,(m_{f'} + m_{f''})$

$f(\pm1)=0,\quad f'(\pm1)=0$

$\text{Rolle: } \exists\, c\in(-1,1)\ \text{with}\ f'(c)=0$

$m_{f'} \ge 3 \qquad (-1,\ c,\ 1)$

$\text{Rolle on } f':\ \exists\, d_1\in(-1,c),\ d_2\in(c,1):\ f''(d_1)=f''(d_2)=0$

$m_{f''} \ge 2$

$m_{f'} + m_{f''} \ge 3 + 2 = 5$

$f = (x^2-1)^2:\quad f' = 4x(x^2-1)\ \Rightarrow\ \text{roots } -1,\ 0,\ 1\ \ (3)$

$f'' = 4(3x^2-1)\ \Rightarrow\ \text{roots } \pm\tfrac{1}{\sqrt3}\ \ (2)$

$\min\,(m_{f'} + m_{f''}) = 5$

$\boxed{5}$