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JEE Main202429 Jan 2024Evening ShiftMathematicsDifferentiationActual

Let y = log e 1 - x 2 1 + x 2 , - 1 < x < 1 . Then at x = 1 2 , the value of 225 y ' - y " is equal to

Options

  1. A732
  2. B746
  3. C742
  4. D736

Correct answer

D. 736

Step-by-step solution

Let, f x = log 1 - x 2 1 + x 2 ⇒ f x = log 1 - x 2 - log 1 + x 2 ⇒ f ' x = - 2 x 1 - x 2 - 2 x 1 + x 2 ⇒ f ' x = - 2 x 1 + x 2 + 1 - x 2 1 - x 4 ⇒ f ' x = 4 x x 4 - 1 ⇒ f " x = x 4 - 1 4 - 4 x 4 x 3 x 4 - 1 2 ⇒ f " x = 4 - 3 x 4 - 1 x 4 - 1 2 ⇒ 225 f ' x - f " x = 225 4 x x 4 - 1 - 4 - 3 x 4 - 1 x 4 - 1 2 Putting, x = 1 2 ⇒ 225 f ' 1 2 - f " 1 2 = 225 2 - 15 16 - 4 - 3 16 - 1 1 16 - 1 2 ⇒ 225 f ' 1 2 - f " 1 2 = 225 - 32 15 - 4 - 19 16 - 15 16 2 ⇒ 225 f ' 1 2 - f " 1 2 = 225 - 32 15 + 4 × 19 × 16 225 ⇒ 225 f ' 1 2

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