For a polynomial $$g(x)$$ with real coefficient, let mg denote the number of distinct real roots of $$g(x)$$ . Suppose S is the set of polynomials with real coefficient defined by $$S=x2$$−$$12a0+a1x+a2x2+a3x3$$:$$a0$$,$$a1$$,$$a2$$,$$a3$$∈RFor a polynomial f, let f ' and f" denote its first and second order derivatives, respectively. Then the minimum possible value of mf'$$+mf$$", where f∈S, is

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For a polynomial $$g(x)$$ with real coefficient, let mg denote the number of distinct real roots of $$g(x)$$ . Suppose S is the set of polynomials with real coefficient defined by $$S=x2$$−$$12a0+a1x+a2x2+a3x3$$:$$a0$$,$$a1$$,$$a2$$,$$a3$$∈RFor a polynomial f, let f ' and f" denote its first and second order derivatives, respectively. Then the minimum possible value of mf'$$+mf$$", where f∈S, is

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$$f(x)=x2$$−$$12h(x)$$;$$h(x)=a0+a1x+a2x2+a3x3$$ Now, $$f(1)=f($$−$$1)=0$$⇒ f′$$($$α$$)=0$$,α∈$$($$−$$1$$,$$1)$$  [Rolle's Theorem] now  f'$$x=2x2$$−$$12x$$ $$h(x)+x2$$−$$12h$$'(x)  Also, f′$$(1)=f$$′$$($$−$$1)=0$$⇒f′$$(x)=0$$ has atleast $$3$$ root, −$$1$$,α,$$1$$ with −$$1$$<α<$$1$$⇒ f′′$$(x)=0$$ will have at least $$2$$ root, say β,γ such that −$$1$$<β<α<γ<$$1$$  [Rolle's Theorem]  So, minmf′′$$=2$$ and we find mf′$$+mf$$"$$=3+2=5$$ Thus, Ans. $$5$$

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