Question

The maximum value of ln⁡xx is:

• A. (A) e
• B. (B) 1e
• C. (C) 2e
• D. (D) 1

Answer 1

Answer: (B)

(B) 1e

Answer 2

Explanation:
Given: Let f(x)=ln⁡xxDifferentiating both sides, we get:f′(x)=ddx(ln⁡xx)=1x(x)−1(ln⁡x)x2⇒f′(x)=1−ln⁡xx2f′′(x)=ddx(f′(x))=ddx(1−ln⁡xx2)=x2ddx(1−ln⁡x)−(1−ln⁡x)ddx(x2)(x2)2=x2(−1x)−(1−ln⁡x)(2x)x4⇒−x−2x+2x(ln⁡x)x4⇒x(−3+2(ln⁡x))x4⇒f′′(x)=−(3−2ln⁡x)x3To find the value of xf′(x)=0⇒1−ln⁡xx2=0⇒1−ln⁡x=0⇒ln⁡x=1⇒x=e1=e(lna⁡b=c⇒b=ac)Now, at x=e,f′′(e)=−(3−2ln⁡e)e3=−3+2e3=−1e3<0At x=e, maximum value of f(x) obtain∴f(x=e)=ln⁡xx=ln⁡ee=1e(∵ln⁡e=1)Hence, the correct option is (B).