Let f x be a polynomial function such that f x + f ' x + f ' ' x = x 5 + 64 . Then, the value of lim x → 1 f x x - 1 is equal to

Step-by-step solution

Let f x = x - 1 x 4 + a x 3 + b x 2 + c x + d f x + f ' x + f '' x = x 5 + a + 4 x 4 + b + 3 a + 16 x 3 + 9 a + 2 b + c - 12 x 2 + d + c + 4 b - 6 a x + c - 2 b ∵     f x + f ' x + f '' x = x 5 + 64 ⇒    a + 4 = 0 ,   b + 3 a + 16 = 0 ,   9 a + 2 b + c - 12 = 0 ,   d + c + 4 b - 6 a = 0 ⇒    a = - 4 ,   b = - 4 ,   c = 56 ,   d = - 64 So lim x → 1 f x x − 1 = lim x → 1 x 4 − 4 x 3 − 4