Minimum number of roots of a polynomial 
For a polynomial () with real coefficients, let $_$ denote the number of distinct real roots of
$()$. Suppose  is the set of polynomials with real coefficients defined by $ =\{{(^2 − 1)^2(_0 +_1+_2^2 + _3^3) ∶ _0, _1, _2, _3 ∈ ℝ}\}$.For a polynomial , let ′and ′′ denote its first and second order derivatives, respectively. Then the minimum possible value of $(_{′} + _{′′})$, where  ∈ , is _____

I know that roots cannot be solved individually but can be solved/found buy the application of Rolle's theorem. How do we find the minimum number of roots for the first and second derevative

As $S={(x-1)}^2{(x+1)}^2(p(x))$ we have $(1,-1)$ are repeated roots!:$f'(1)=f'(-1)=0$
$f(1)=f(-1)=0$ hence $f'(k)=0$ for some $k$ in $(-1,1)$.Also $f'(-1)=f'(1)=0$(repeated root).
$f'(-1)=f'(k)=0$ hence there is some $d$ in $(-1,k)$ $f''(d)=0$ similarly threre is some $m$ in $(k,1) $ for $f''(m)=0$.
thus minimum number of roots of $f'$ is $3$: roots are $(-1,1,k)$
and for $f''$ roots are :$(m,d)$
Thus a minimum value is 5
