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

$f(x)=\left(x^{2}-1\right)^{2} \cdot p(x)$
where $p(x)=a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}$
$\because f(x)$ has two repeated roots $x=1$ and $x=-1$
So $f^{\prime}(x)$ has at least 3 roots $x=1, x=-1$ and $x=c$
Where $c \in(-1,1)$, so $m_{f}=3$
Between any two distinct roots of $f^{\prime}(x)$ there will be at least one root of
$f^{\prime \prime}(x)$, so $m_{f^{\prime \prime}}=2$
$m_{f^{\prime}}+m_{f^{\prime \prime}}=5$