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JEE Main20231 Feb 2023Evening ShiftMathematicsDifferentiationActual

If y x = x x , x > 0 , then y " 2 - 2 y ' 2 is equal to :

Options

  1. A8 log e 2 - 2
  2. B4 log e 2 + 2
  3. C4 log e 2 2 - 2
  4. D4 log e 2 2 + 2

Correct answer

C. 4 log e 2 2 - 2

Step-by-step solution

Given, y = x x ⇒ ln y = x ln x Differentiate both sides w.r.t. x 1 y . y ' = 1 + ln x ⇒ y ' = y 1 + l n x   . . . . . . . ( 1 ) ⇒ y ' = x x 1 + l n x at x = 2 ⇒ y = 4 So, y ' ( 2 ) = 4 1 + l n 2   . . . . . . . ( 2 ) Now differentiate the equation 1 , y " = y ' 1 + ln x + y x ⇒ y " ( 2 ) = y ' 1 + ln 2 + 2 ⇒ y " ( 2 ) - y ' ( 2 ) = y ' ( ln 2 ) + 2 ⇒ y " ( 2 ) - 2 y ' ( 2 ) = ln 2 - 1 y ' ( 2 ) + 2 = 4 ( ln 2

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