Write the maximum value of f x = x 1 x .

Given : f ( x ) = x 1 x Taking log on both sides, we get log e f ( x ) = 1 x log e x Differentiating with respect to x , we get 1 f ( x ) f ' ( x ) = - 1 x 2 log e x + 1 x 2 ⇒ f ' ( x ) = f ( x ) 1 x 2 ( 1 - log e x ) ⇒ f ' ( x ) = x 1 x 1 x 2 - 1 x 2 log e x ⇒ f ' ( x ) = x 1 x - 2 ( 1 - log e x ) For local maxima or local minima, we must have f ' ( x ) = 0 ⇒ x 1 x - 2 ( 1 - log x ) = 0 ⇒ log x = 1 ⇒ x = e Now, f ' ' ( x ) = d d x x 1 x 1 x 2 - 1 x 2 log e x = x 1 x 1 x 2 - 1 x 2 log x 2 + x 1 x - 2 x 3 + 2 x 3 log x - 1 x 3 ⇒ f ' ' ( x ) = x 1 x 1 x 2 - 1 x 2 log x 2 + x 1 x - 3 x 3 + 2 x 3 log x At x = e f ' ' ( e ) = e 1 e 1 e 2 - 1 e 2 log e 2 + e 1 e - 3 e 3 + 2 e 3 log e = - e 1 e 1 e 3 < 0 So, x = e is a point of local maximum. Thus, the maximum value is given by f ( e ) = e 1 e

Write the maximum value of f ( x ) = log x x , if it exists.

Given : f ( x ) = log x x ⇒ f ' ( x ) = d d x log x x = 1 - log x x 2 For local maxima or local minima, we must have f ' ( x ) = 0 ⇒ 1 - log x x 2 = 0 ⇒ 1 - log x = 0 ⇒ log x = 1 ⇒ log x = log e ⇒ x = e Now, f ' ' ( x ) = d d x 1 - log x x 2 = - x - 2 x ( 1 - log x ) x 4 = - 3 x + 2 x log x x 4 At x = e , f ' ' ( e ) = - 3 e + 2 e log e e 4 = - 1 e 3 < 0 So, x = e is a point of local maximum. Thus, the local maximum value is given by f ( e ) = log e e = 1 e