Let f x be a polynomial of degree four and having its extreme values at x = 1 and x = 2 . If lim x → 0 1 + f x x 2 = 3 , then f 2 is equal to

Step-by-step solution

Let f x = a x 4 + b x 3 + c x 2 + d x + e Then lim x → 0 1 + f x x 2 = 3 ⇒ lim x → 0 1 + a x 2 + b x + c + d x + e x 2 = 3 This limit exists when d = e = 0 . So, lim x → 0 1 + a x 2 + b x + c = 3 ⇒ c + 1 = 3 ⇒ c = 2 It is given, x = 1 and x = 2 are the solution of f ' x = 0 . ⇒ f ' x = 4 a x 3 + 3 b x 2 + 2 c x ⇒ x 4 a x 2 + 3 b x + 2 c = 0 Given 1 ,   2 are the roots of the quadratic equation. ⇒ Sum of roots = - 3 b 4 a = 1 + 2 = 3 ⇒ b = - 4 a Produc