JEE Advanced 2020 — Paper 1 — Question 44

For a polynomial $g(x)$ with real coefficients, let $\mathrm{m}_{\mathrm{g}}$ denote the number of distinct real roots of $\mathrm{g}(\mathrm{x})$. 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 $f^{\prime \prime}$ denote its first and second order derivatives, respectively. Then the minimum possible value of $\left(\mathrm{m}_{\mathrm{f}^{\prime}}+\mathrm{m}_{\mathrm{f}^{\prime \prime}}\right)$, where $\mathrm{f} \in \mathrm{S}$, is $\_\_\_\_$