Let f be any continuous function on 0 , 2 and twice differentiable on 0 , 2 . If f 0 = 0 , f 1 = 1 and f 2 = 2 , then :

Step-by-step solution

Given f 0 = 0 ,   f 1 = 1 and f 2 = 2 Let, h x = f x - x Clearly h x , will be continuous and twice differentiable on 0 ,   2 . Now, h 0 = h 1 = h 2 = 0 By Rolle's mean value theorem in 0 ,   1 , we get h ' c 1 = 0       ⇒ f ' c 1 - 1 = 0       ⇒ f ' c 1 = 1 , where c 1 ∈ 0 ,   1 Also, in the interval 1 ,   2 h ' c 2 = 0 ⇒ f ׀ c 2 - 1 = 0 ⇒ f ' c 2 = 1 , where c 2 ∈ 1 ,   2 Now, use Rolle&#3