Let f x be a polynomial of degree 6 in x , in which the coefficient of x 6 is unity and it has extrema at x = - 1 and x = 1 . If lim x → 0 f x x 3 = 1 , then 5 · f 2 is equal to

Step-by-step solution

Let f x = x 6 + a x 5 + b x 4 + c x 3 + d x 2 + e x + f as lim x → 0 f x x 3 = 1 non-zero finite So, d = e = f = 0 and f x = x 3 x 3 + a x 2 + b x + c Hence, lim x → 0 f x x 3 = c = 1 Now, as f x = x 6 + a x 5 + b x 4 + x 3 and f ' x = 0 at x = 1 and x = - 1 i.e., f ' x = 6 x 5 + 5 a x 4 + 4 b x 3 + 3 x 2 f ' 1 = 0 ⇒ 6 + 5 a + 4 b + 3 = 0 ⇒ 5 a + 4 b = - 9 and f ' - 1 = 0 ⇒ - 6 + 5 a - 4 b + 3 = 0 ⇒ 5 a - 4 b = 3 Solving both we get, a = - 6 10 = - 3 5 ;   b