Let c , k ∈ R . If f x = c + 1 x 2 + 1 - c 2 x + 2 k and f x + y = f x + f y - x y , for all x , y ∈ R , then the value of 2 f 1 + f 2 + f 3 + … … + f 20 is equal to ______.

Step-by-step solution

Given f x = c + 1 x 2 + 1 - c 2 x + 2 k           ⋯ i & f x + y = f x + f y - x y    ∀ x y ∈ R So f ' x = 2 c + 1 x + 1 - c 2 Now lim y → 0 f x + y - f x y = lim y → 0 f y - x y y = lim y → 0 f y y - x ⇒ f ' x = f ' 0 - x Integrating, we get f x = - x 2 2 + f ' 0 x + λ but f 0 = 0 ⇒ λ = 0 i.e. f x = - x 2 2 + 1 - c 2 x           ⋯ i i ∴ as f ' 0 = 1 - c 2 Now comparing equation i and