JEE Advanced 2017 · Paper 1 · Mathematics Q16

MathematicsSingle correct+3 / -1Question 52 of 54

Passage

Answer Q.52, Q.53 and Q.54 by appropriately matching the information given in the three columns of the following table.

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 $f'(x)$ .

Which of the following options is the only CORRECT combination?

(I) (i) (P)

(II) (ii) (Q)

(III) (iii) (R)

(IV) (iv) (S)

Answer and worked solution

Official answer: B

Derivatives. $f(x)=x+\ln x-x\ln x$ , so
$f'(x)=\frac1x-\ln x,\qquad f''(x)=-\frac1{x^2}-\frac1x<0.$

What holds.

$f'$ is decreasing everywhere, with $f'(1)=1>0$ and $f'(e)=\tfrac1e-1<0$ . So $f'$ has one zero, in $(1,e)$ . (II) is true; (III) is false; (IV) is false since $f''<0$ .
$f$ increases before that zero and decreases after it. So (P) is true (increasing on $(0,1)$ ) and (Q) is true (decreasing on $(e,e^2)$ ).
(R) is false; (S) is true.
$f(1)=1>0$ and $f(e^2)=2-e^2<0$ , so (I) is true.
$f=x(1-\ln x)+\ln x\to-\infty$ : (ii) is true and (i) is false.
$f'\to-\infty$ : (iii) is true. $f''\to0$ : (iv) is true.

This question. (II)(ii)(Q) is all true. Each of the other options contains a false part.

(B)

Topics:Monotonicity Limits

JEE Advanced 2017 · Paper 1, Mathematics question 16