Let f : R → R be a function satisfying f x + y = f x + 2 y 2 + k x y for all x , y ∈ R . If f 1 = 2 and f 2 = 8 , then f x is equal to

Step-by-step solution

We have, f x + y = f x + 2 y 2 + k x y for all x , y ∈ R ⇒ f x + y - f x y = 2 y + k x for all x ∈ R ⇒ lim y → 0 f x + y - f x y = lim y → 0 2 y + k x ⇒ f ' x = k x for all x ∈ R ⇒ f x = k x 2 2 + C for all x ∈ R [by integration] But f 1 = 2 and f 2 = 8 ∴   2 = k 2 + C and 8 = 2 k + C k = 4 and C = 0 Hence, f x = 2 x 2 for all x ∈ R So, option (a) is correct.