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KCET2016MathematicsApplication of Derivatives

The maximum value of ( ( 1 x )^ x ) is

Options

  1. A( e )
  2. Be^ e
  3. C( e^ 1 e )
  4. D( ( 1 e )^ e )

Correct answer

C. ( e^ 1 e )

Step-by-step solution

Let ( y= ( 1 x )^ x ) Taking log on both the sides, we get [ array l y=x ( 1 x )=-x x y=-x x (1) array ] Now, differentiating Eq. (1) with respect to ( x ), we get [ array l 1 y d y d x =- (x 1 x + x ) 1 y d y d x =-(1+ x) d y d x =-y(1+ x) (2) d y d x =- ( 1 x )^ x (1+ x) array ] [ array l Now, d y d x =0 -y(1+ x)=0 x=-1 x=e⁻¹ array ] Now, differentiating Eq. (2) with respect to ( x ), we get [ array l d² y d x² =-(1+ x) d y d x - y x d² y d x² =y(1+ x)²- y x d² y d x² = ( 1 x )^ x (1+ x)²- ( 1 x )^ x 1 x At x=e⁻¹

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