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 $$S = \left\{ \left( x ^ { 2 } - 1 \right) ^ { 2 } \left( a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + a _ { 3 } x ^ { 3 } \right) : a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 } \in \mathbb { R } \right\}$$ For a polynomial $f$, let $f ^ { \prime }$ and $f ^ { \prime \prime }$ denote its first and second order derivatives, respectively. Then the minimum possible value of ( $m _ { f ^ { \prime } } + m _ { f ^ { \prime \prime } }$ ), where $f \in S$, is $\_\_\_\_$
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
$$S = \left\{ \left( x ^ { 2 } - 1 \right) ^ { 2 } \left( a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + a _ { 3 } x ^ { 3 } \right) : a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 } \in \mathbb { R } \right\}$$
For a polynomial $f$, let $f ^ { \prime }$ and $f ^ { \prime \prime }$ denote its first and second order derivatives, respectively. Then the minimum possible value of ( $m _ { f ^ { \prime } } + m _ { f ^ { \prime \prime } }$ ), where $f \in S$, is $\_\_\_\_$