Let S be the set of real numbers p such that there is no non-zero continuous function f : R → R satisfying ∫ 0 x f t d t = p f x for all x ∈ R . Then, S is

Step-by-step solution

Given, ∫ 0 x f t d t = p f x ⇒ f x = p f ' x ⇒ f ' x f x = 1 p ⇒ log f x = x p + C ⇒ f x = A e x / p Putting x = 0 f 0 = A e 0 ⇒ A = 0    ∵ f 0 = 0 ∵ f x = 0 ∴ p f 0 = 0 Case I f 0 = 0 , p ≠ 0 There is no non-zero continuous to f x . Case II p = 0 ∴ ∫ 0 x f t d t = 0 , ∀ x ∈ R If is possible when f x = 0 . Hence, ∀ p ∈ R . There is no non-zero continous function. Hence, S ∈ R .