Let f : ( 0 , ∞ ) → ( 0 , ∞ ) be a differentiable function such that f ( 1 ) = e and lim t → x t 2 f 2 ( x ) - x 2 f 2 ( t ) t - x = 0 . If f ( x ) = 1 , then x is equal to:

Question

Let f : ( 0 , &#8734; ) &#8594; ( 0 , &#8734; ) be a differentiable function such that f ( 1 ) = e and lim t &#8594; x t 2 f 2 ( x ) - x 2 f 2 ( t ) t - x = 0 . If f ( x ) = 1 , then x is equal to:

Quantrex Academy · Accepted Answer

Correct answer: A. 1 e

lim t &#8594; x t 2 f 2 ( x ) - x 2 f 2 ( t ) t - x = 0 Using L'Hospital lim t &#8594; x 2 t f 2 ( x ) - x 2 2 f ( t ) f ' ( t ) 1 = 0 &#8658; 2 x f 2 ( x ) - x 2 2 f ( x ) f ' ( x ) = 0 &#8658; 2 &#160; x &#160; f ( x ) [ &#160; f ( x ) - x f ' ( x ) ] = 0 But f ( x ) &#8800; 0 So, x f ' ( x ) = f ( x ) &#8658; x d y d x = y &#8658; 1 y d y = 1 x d x Integration gives ln y = ln x + ln c &#8658; y = c x &#8658; f ( x ) = c x Now f ( 1 ) = c = e (given) So, f ( x ) = e &#160; x Now if f ( x ) = 1 , then

Let f : ( 0 , ∞ ) → ( 0 , ∞ ) be a differentiable function such that f ( 1 ) = e and lim t → x t 2 f 2 ( x ) - x 2 f 2 ( t ) t - x = 0 . If f ( x ) = 1 , then x is equal to:

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Correct answer

Step-by-step solution

More from Continuity and Differentiability

Let f : ( 0 , &#8734; ) &#8594; ( 0 , &#8734; ) be a differentiable function such that f ( 1 ) = e and lim t &#8594; x t 2 f 2 ( x ) - x 2 f 2 ( t ) t - x = 0 . If f ( x ) = 1 , then x is equal to:

Options

Correct answer

Step-by-step solution

More from Continuity and Differentiability

Let f : ( 0 , ∞ ) → ( 0 , ∞ ) be a differentiable function such that f ( 1 ) = e and lim t → x t 2 f 2 ( x ) - x 2 f 2 ( t ) t - x = 0 . If f ( x ) = 1 , then x is equal to: