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JEE Main202329 Jan 2023Morning ShiftMathematicsFunctionsActual

Let f : R → R be a function such that f x = x 2 + 2 x + 1 x 2 + 1 . Then

Options

  1. Af x is many-one in - ∞ , - 1
  2. Bf x is many-one in 1 , ∞
  3. Cf x is one-one in [ 1 , ∞ ) but not in - ∞ , ∞
  4. Df x is one-one in - ∞ , ∞

Correct answer

C. f x is one-one in [ 1 , ∞ ) but not in - ∞ , ∞

Step-by-step solution

Given, f x = x 2 + 2 x + 1 x 2 + 1 f ' ( x ) = x 2 + 1 ( 2 x + 2 ) - ( x 2 + 2 x + 1 ) 2 x x 2 + 1 2 ⇒ f ' ( x ) = 2 x 3 + 2 x 2 + 2 x + 2 - 2 x 3 - 4 x 2 - 2 x x 2 + 1 2 ⇒ f ' ( x ) = 2 ( 1 - x 2 ) x 2 + 1 2 = - 2 ( x + 1 ) ( x - 1 ) x 2 + 1 2 Clearly f ( x ) is one-one in - ∞ , - 1 and also in ( 1 , ∞ ) but f ( x ) is not one-one in - ∞ , ∞

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