Test Index

Application of Derivatives

Paragraph for next 3 questions

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)

Column 1	Column 2	Column 3
(I) $f(x)=0$ for some $x\in(1,e^2)$	(i) $\lim\limits_{x\to\infty} f(x)=0$	f is increasing in $(0,1)$
(II) $f'(x)=0$ for some $x\in(1,e)$	(ii) $\lim\limits_{x\to\infty} f(x)=-\infty$	(Q) f is decreasing in $(e, e^2)$
(III) $f'(x)=0$ for some $x\in(0,1)$	(iii) $\lim\limits_{x\to\infty} f'(x)=-\infty$	$f'$ increasing in $(0,1)$
(IV) $f''(x)=0$ for some $x\in(1,e)$	(iv) $\lim\limits_{x\to\infty} f''(x)=0$	(S) $f'$ decresing in $(e, e^2)$