Let f(x) be a continuously differentiable function on the interval (0, ) such that f(1)=2 and t x t¹⁰ f(x)-x¹⁰ f(t) t^9-x^9 =1 for each x0 . Then, for all x0, f(x) is equal to

Question

Let f(x) be a continuously differentiable function on the interval (0, ) such that f(1)=2 and t x t¹⁰ f(x)-x¹⁰ f(t) t^9-x^9 =1 for each x>0 . Then, for all x>0, f(x) is equal to

Quantrex Academy · Accepted Answer

Correct answer: B. 9 11 x + 13 11 x¹⁰

aligned & _ t x t¹⁰ f(x)-x¹⁰ f(t) t^9-x^9 =1 & _ t x 10 t^9 f(x)-x¹⁰ f^ (t) 9 t^8 =1 aligned aligned & 10 x f(x)-x^2 f^ (x)=9 & x^2 f^ (x)=10 x f(x)-9 & f^ (x)= 10 f(x) x - 9 x^2 & d y d x - 10 x y=- 9 x^2 & y 1 x¹⁰ = - 9 x^2 1 x¹⁰ d x & y x¹⁰ = 9 11 x¹¹ +c...(1) aligned aligned & f (1)=2 2 1 = 9 11 + c c = 13 11 & f ( x )= 9 11 x + 13 11 x ¹⁰ aligned Option (2) is correct.

Let f(x) be a continuously differentiable function on the interval (0, ) such that f(1)=2 and t x t¹⁰ f(x)-x¹⁰ f(t) t^9-x^9 =1 for each x>0 . Then, for all x>0, f(x) is equal to

Options

Correct answer

Step-by-step solution

More from Limits