99 Percentile Qs Bank for JEE MainMathematicsApplication of Derivatives

The function f ( x ) = x 1 x is:

Options

Aincreasing in ( 1 ,   ∞ )
Bdecreasing in ( 1 ,   ∞ )
Cincreasing in ( 1 ,   e ) , decreasing in ( e ,   ∞ )
Ddecreasing in ( 1 ,   e ) , increasing in ( e ,   ∞ )

Correct answer

C. increasing in ( 1 ,   e ) , decreasing in ( e ,   ∞ )

Step-by-step solution

Let y = x 1 x ⇒ log y = 1 x log x ⇒ 1 y d y d x = 1 x 2 - log e x x 2 ⇒ d y d x = x 1 x 1 - log e x x 2 For 1 < x <∞,   x 1 x > 0 and 1 - log e x x 2 > 0   in   1 ,   e and 1 - log e x x 2 < 0   in   e ,   ∞ Hence, f ( x ) is increasing in ( 1 ,   e ) and decreasing in ( e ,   ∞ ) .

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