Let f [ 1 , ∞ ) → [ 2 , ∞ ) be a differentiable function such that f ( 1 ) = 1 3 . If 6 ∫ 1 x f t dt = 3 x f x - x 3 for all x ≥ 1 , then the value of 3 f 2 is

Step-by-step solution

Given, f 1 = 1 3 and 6 ∫ 1 x f ⁡ t dt = 3 x f ⁡ x - x 3 for all x ≥ 1 Using Newton- Leibnitz formula. Differentiating both sides, ⇒ 6 f ⁡ x · 1 - 0 = 3 f x + 3 x f ′ x - 3 x 2 ⇒ 3 x f ′ x - 3 f x = 3 x 2 ⇒ f ′ x - 1 x f x = x ⇒ x f ′ x - f x x 2 = 1 ⇒ d dx f x x = 1 Integrating both sides, ⇒ f x x = x + C ∵ f 1 = 1 3 1 3 = 1 + C ⇒ C = - 2 3 f x = x 2 - 2 3 x ⇒ f 2 = 4 - 4 3 = 8 3 ∴ 3 f 2 = 8