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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 INCORRECT combination?
Differentiation and its Application
Options for question 1, choose one

Solution

Concept. Here $f(x)=x+\ln x-x\ln x$ on $(0,\infty)$. Mark each entry true or false; the incorrect combination contains a false entry. Formulas used. (1) $f'(x)=\frac1x-\ln x$, $f''(x)=-\frac1{x^2}-\frac1x<0$Step 1 - True entries. (I), (II), (ii), (iii), (iv), (P), (Q), (S) are true (see the checks: $f(1)=1$, $f(e^2)<0$; $f'(1)>0>f'(e)$; $f\to-\infty$, $f'\to-\infty$, $f''\to0$; $f'>0$ on $(0,1)$, $f'<0$ on $(e,e^2)$, $f''<0$). Step 2 - False entries. (III) ($f'>0$ on $(0,1)$), (IV) ($f''<0$), (i) ($f\to-\infty$, not $0$), (R) ($f'$ is decreasing). Step 3 - Options. (A) (I)(iii)(P), (B) (II)(iv)(Q), (D) (II)(iii)(P) are made of true entries. (C) (III)(i)(R) is made of false entries: it is the incorrect one. Answer: (C)

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