JEE Advanced 2023 Paper 1 · Mathematics · Differentiation and Application of Derivatives · medium

Let h(x)=(ln x)/(x) for x>0. Using properties of h, decide which of the following statements is/are correct.

One or more options correct

Let

h(x)=\dfrac{\ln x}{x}

for

x>0

. Using properties of

h

, decide which of the following statements is/are correct.

Show answer and solution

Answer: A, B, C

h'(x)=\dfrac{1-\ln x}{x^2}

, so

h

increases on

(0,e)

, attains its maximum at

x=e

and decreases on

(e,\infty)

. Also

a^b>b^a\iff h(a)>h(b)

for

a,b>1

, because

b\ln a>a\ln b

is the same as

\frac{\ln a}{a}>\frac{\ln b}{b}

.
(A) True: maximum at

x=e

h(e)=\frac1e

.
(B) True:

h(e)>h(\pi)

gives

\frac{1}{e}>\frac{\ln\pi}{\pi}

, i.e.

\pi>e\ln\pi

, i.e.

e^\pi>\pi^e

(equivalently

\pi^e<e^\pi

).
(C) True:

h(2)=\frac{\ln2}{2}=\frac{2\ln2}{4}=h(4)

, and indeed

2^4=4^2=16

.
(D) False:

2023,2024>e

lie on the decreasing branch, so

h(2023)>h(2024)

, which means

2023^{2024}>2024^{2023}

, the opposite of the stated inequality.

Original question written in the JEE Advanced 2023 Paper 1 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake