MHT CET2018MathematicsApplication of Derivatives

The minimum value of the function f x = x log ⁡ x is

Options

A- 1 e
B- e
C1 e
De

Correct answer

A. - 1 e

Step-by-step solution

f ( x ) = x log x ⇒ f ' ( x ) = x × 1 x + log x × 1 ⇒ f ' ( x ) = 1 + log x Now for f(x) to be minimum, f ' ( x ) = 0 ⇒ 1 + log x = 0 ⇒ log e x = − 1 ⇒ x = e − 1 = 1 e Also f " ( x ) = 1 x ⇒ f " ( 1 e ) = 1 1 e = 3 > 0 ⇒ f ( x ) is minimum at x = 1 e and the minimum value is f ( 1 e ) = 1 e log 1 e = − 1 e

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