Let f be a differentiable function satisfying f x = 2 3 ∫ 0 3 f λ 2 x 3 d λ , x > 0 and f 1 = 3 . If y = f x passes through the point α , 6 , then α is equal to _______.

Step-by-step solution

Given, f x = 2 3 ∫ 0 3 f λ 2 x 3 d λ Now let λ 2 x 3 = t ⇒ 2 λ x 3   d λ = d t ⇒ d λ = 3 2 · 1 x x · 3 t d t ⇒ d λ = 3 2 · 1 x · d t t So, f x = 1 x ∫ 0 x f t t d t x f x = ∫ 0 x f t t d t Now differentiating both side we get, ⇒ x · f ' x + f x 2 x = f x x ⇒ x · f ' x = f x 2 x ⇒ d y y = d x 2 x Now integrating both side we get, ⇒ ∫ d y y = ∫ d x 2 x ⇒ ln y = 1