Question

For a polynomial $g\left(x\right)$ with real coefficients, let $m_g$ denote the number of distinct real roots of $g\left(x\right)$. Suppose $S$ is the set of polynomial with real coefficients defined by
$S=\left[\right(x^2-1{)}^2(a_0+a_{1}x+a_2{x}^2+a_3{x}^3):a_0,a_1,a_2,a_3\in R]$
For a polynomial $f$, let $f^{\prime }$ and $f^{\prime \prime }$ denote first and second order derivatives, resepectively. Then the minimum possible value of $(m_{f^{\prime }}+m_{f^{\prime \prime }}),$ where $f\in S$, is

Answer 1

$f\left(x\right)=(x^2-1{)}^{2}h(x)$; $h\left(x\right)=a_0+a_{1}x+a_2{x}^2+a_3{x}^3$

Now, $f\left(1\right)=f(-1)=0$

$\Rightarrow f^{\prime }\left(\alpha \right)=0,\alpha \in (-1,1)$
[Rolle's Theorem]

Also, $f^{\prime }\left(1\right)=f^{\prime }(-1)=0$
$\Rightarrow f^{\prime }\left(x\right)$ has atleast $3$ roots $-1,\alpha ,1$ with $-1<\alpha <1$

$f^{\prime \prime }\left(x\right)=0$ will have atleast 2 roots, say $\beta ,\gamma$ such that $-1<\beta <\alpha <\gamma <1$
[Rolle's Theorem]

So, $min\left(m_{f^{\prime \prime }}\right)=2$

and we find $m_{f^{\prime }}+m_{f^{\prime \prime }}=5$ for $f\left(x\right)=(x^2-1{)}^{2}h(x)$