For a polynomial g (x) with real coefficients, let m _ g denote the...

$f$\left(x\right)$is a polynomial in$x$such that$f\left(x\right) = \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\}$</p><p>So,${f}^{'}\left(x\right) = \left({x}^{2} - 1\right)\left[q\left(x\right)\right]$where$q$\left(x\right)$ is polynomial roots of ${f}^{′}\left(x\right) = - 1, 1$

So, by Rolle’s theorem, $f\left(1\right) = 0 = f\left(- 1\right) ⇒ f ' \left(\alpha\right) = 0$, where $\alpha \in \left(- 1, 1\right)$

$⇒$ minimum $m{f}^{′} = 3$

Again by Rolle’s theorem, at least one real root lies in \left(- 1, \alpha\right)$and at least one real root lies in$\left(\alpha, 1\right)$of${f}^{″}\left(x\right)$</p><p>So, minimum$m{f}^{″} = 2$<br />$⇒$Minimum possible value of \left(m{f}^{′} + m{f}^{″}\right) = 5$