Functions

For a polynomial g(x) with real coefficients, let mg denote the number of distinct real roots of g(x) Suppose S is the set of polynomials with real coefficients defined by
S={(x2−1)2(a0+a1x+a2x2 +a3x3:a0,a1,a2,a3∈R}
For a polynomial f, let f' and f' denote its first and second order derivatives, respectively. Then theminimum possible value of (mf,+mf",), where f∈S, is _______.