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 polynomials 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 ∈ R . For a polynomial f , let f ' and f ″ denote its first and second order derivatives respectively. Then the minimum possible value of m f ′ + m f

Step-by-step solution

f x is a polynomial in x such that f x = 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 ∈ R So, f ' x = x 2 − 1   q x where q x is polynomial roots of f ′ x = − 1 ,   1 So, by Rolle’s theorem, f 1 = 0 = f − 1 ⇒ f ' α = 0 , where α ∈ − 1 ,   1 ⇒ minimum m f ′ = 3 Again by Rolle’s theorem, at least one real root lies in − 1 ,   α an