For a polynomial g(x) with real coefficients, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomial with real coefficients defined by S=[(x2−1)2(a0+a1x+a2x2+a3x3):a0,a1,a2,a3∈R]For a polynomial f, let f′ and f′′ denote first and second order derivatives, resepectively. Then the minimum possible value of (mf′+mf′′), where f∈S, is

For a polynomial g(x) with real coefficients, let mg denote the number

1.	For a polynomial g(x) with real coefficients, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomial with real coefficients defined by S=[(x2−1)2(a0+a1x+a2x2+a3x3):a0,a1,a2,a3∈R]For a polynomial f, let f′ and f′′ denote first and second order derivatives, resepectively. Then the minimum possible value of (mf′+mf′′), where f∈S, is
Answer» For a polynomial $g (x)$ with real coefficients, let $m_{g}$ denote the number of distinct real roots of $g (x)$ . Suppose $S$ is the set of polynomial with real coefficients defined by $S = [(x^{2} - 1)^{2} (a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3}) : a_{0}, a_{1}, a_{2}, a_{3} \in R]$ For a polynomial $f$ , let $f^{'}$ and $f^{''}$ denote first and second order derivatives, resepectively. Then the minimum possible value of $(m_{f^{'}} + m_{f^{''}}),$ where $f \in S$ , is