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 $f^{\prime }$ and $f"$ denote its first and second order derivatives, respectively. Then the minimum possible value of $\left(m_{f^{\prime }}+m_{f^{\prime \prime }}\right)$, where $f\in S$, is

Answer 1

Answer: 5

$f(x)={\left(x^2-1\right)}^2\cdot p(x)$
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 $m_{f^{\prime \prime }}=2$
$m_{f^{\prime }}+m_{f^{\prime \prime }}=5$