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JEE Main202227 Jun 2022Evening ShiftMathematicsDifferentiationActual

If y x = x x x , x > 0 then d 2 x d y 2 + 20 at x = 1 is equal to

Correct answer

0

Step-by-step solution

Given, y x = x x x Taking log e both side ln y x = x 2 · ln x Now differentiating both side w.r.t x we get, 1 y x · y ' x = x 2 x + 2 x · ln x y ' x = y x x + 2 x   ln x .......(i) Given y 1 = 1 , so y ' 1 = 1 Now rewriting equation (i) again we get, d x d y = 1 x x 2 + 1 1 + 2 ln x Now d 2 x d y 2 = d d x x x 2 + 1 1 + 2 ln x - 1 d x d y ⇒ d 2 x d y 2 = - x x 2 1 + 2 ln x x 2 + 3 + 2 x 2 ln x x x 2 1 + 2 ln x 3 × 1 ⇒ d 2 x d y 2 x = 1 = - 4 So, d 2 x d y 2 x = 1 + 20 =

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