Suppose a continuous function f : 0 , ∞ → R satisfies f x = 2 ∫ 0 x t f t d t + 1 , ∀ x ≥ 0 . Then, f 1 equals

Step-by-step solution

It is given that a continuous function f : [ 0 , ∞ ) ⟶ R satisfies f x = 2 ∫ 0 x t f t d t + 1 , ∀ x ≥ 0 ∴   f ' x = 2 x f x ⇒   f ' x f x = 2 x On integrating both sides, we get In | f x | = x 2 + c ⇒   f x = K e x 2 , where K = e c ∵   f 0 = 2 ∫ 0 0 t f t d t + 1 = 1 ∴   K = 1 ∴   f 1 = e