Let f x = lim r → x 2 r 2 ( f ( r ) ) 2 - f ( x ) f ( r ) r 2 - x 2 - r 3 e f ( r ) r be differentiable in ( - ∞ , 0 ) ∪ ( 0 , ∞ ) and f ( 1 ) = 1 . Then the value of a e , such that f ( a ) = 0 , is equal to ______.

Step-by-step solution

Given, f ( 1 ) = 1 , f ( a ) = 0 And f x = lim r → x 2 r 2 ( f ( r ) ) 2 - f ( x ) f ( r ) r 2 - x 2 - r 3 e f ( r ) r ⇒ f 2 ( x ) = lim r → x 2 r 2 f 2 ( r ) - f ( x ) f ( r ) r 2 - x 2 - r 3 e f ( r ) r ⇒ f 2 ( x ) = lim r → x 2 r 2 f ( r ) r + x ( f ( r ) - f ( x ) ) r - x - r 3 e f ( r ) r ⇒ f 2 ( x ) = 2 x 2 f ( x ) x + x lim r → x ( f ( r ) - f ( x ) ) r - x - x 3 e f ( x ) x ⇒ f 2 ( x ) = 2 x 2 f ( x ) 2 x f ' ( x ) - x 3 e f ( x ) x Now, taking f x = y we get, ⇒ y 2 = x y d y d x - x 3 e y x ⇒ y x = d y d x