Question

For a polynomial $g(x)$ with real coefficients, let $m_{g}$ denote the number of distinct real roots of $g(x)$. Suppose $S$ in 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 R \right\\}
For a polynomial $f$, let $f'$ and $f"$ denote its first and second order derivatives, respectively. Then the minimum possible value of $\left(m_{f'}+m_{f''}\right)$, where $f \in S$, is

Answer 1

Then the minimum possible value of $\left(m_{f'}+m_{f''}\right)$, where $f \in S$, is $\underline{5}.$