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Yukti Vault

Differentiation and its Application - JEE Mathematics question (JEE Advanced 2017 Paper 1)

Comprehension · Questions 1–1

Read the passage given below and answer the questions that follow.

Let $f(x)=x+\log_e x-x\log_e x$, $x\in(0,\infty)$. Column 1 contains information about zeros of $f(x)$, $f'(x)$ and $f''(x)$. Column 2 contains information about the limiting behavior of $f(x)$, $f'(x)$ and $f''(x)$ at infinity. Column 3 contains information about increasing/decreasing nature of $f(x)$ and $f'(x)$. Table (Column 1 | Column 2 | Column 3): (I) $f(x)=0$ for some $x\in(1,e^2)$ | (i) $\lim_{x\to\infty}f(x)=0$ | (P) $f$ is increasing in $(0,1)$; (II) $f'(x)=0$ for some $x\in(1,e)$ | (ii) $\lim_{x\to\infty}f(x)=-\infty$ | (Q) $f$ is decreasing in $(e,e^2)$; (III) $f'(x)=0$ for some $x\in(0,1)$ | (iii) $\lim_{x\to\infty}f'(x)=-\infty$ | (R) $f'$ is increasing in $(0,1)$; (IV) $f''(x)=0$ for some $x\in(1,e)$ | (iv) $\lim_{x\to\infty}f''(x)=0$ | (S) $f'$ is decreasing in $(e,e^2)$.
JEE Advanced 2017 Paper 1Question 1
Which of the following options is the only CORRECT combination?
Differentiation and its Application
Options for question 1, choose one

Solution

Concept. Here $f(x)=x+\ln x-x\ln x$ on $(0,\infty)$. Find $f'$ and $f''$, then test every entry of the three columns once. Formulas used. (1) $f'(x)=\frac1x-\ln x$, $f''(x)=-\frac1{x^2}-\frac1x<0$Step 1 - Column 1. (I) $f(1)=1>0$, $f(e^2)=2-e^2<0$: true. (II) $f'(1)=1>0$, $f'(e)=\frac1e-1<0$: true. (III) On $(0,1)$, $\frac1x>0$ and $-\ln x>0$, so $f'>0$: false. (IV) $f''<0$ always: false. Step 2 - Column 2. $f=x(1-\ln x)+\ln x\to-\infty$: (i) false, (ii) true. $f'\to-\infty$: (iii) true. $f''\to0$: (iv) true. Step 3 - Column 3. (P) $f'>0$ on $(0,1)$: true. (Q) on $(e,e^2)$, $f'<\frac1e-1<0$: true. (R) $f''<0$, so $f'$ decreases: false. (S) true. Step 4 - Options. (A) has (i), false. (C) has (III), false. (D) has (IV), false. (B) (II)(ii)(Q): all true. Answer: (B)

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