Let $f\left(x\right)=x+{\log }_{e}x-x{\log }_{\mathrm{e}}x,x\in \left(0, \infty \right)$
$•$   Column 1 contains information about zeros of $f\left(x\right)$ , $f\text{'}\left(x\right)$ and $f\text{'}\text{'}\left(x\right)$
$•$   Column 2 contains information about the limiting behaviour of $f\left(x\right)$ , $f\text{'}\left(x\right)$ and $f\text{'}\text{'}\left(x\right)$ at infinity.
$•$   Column 3 contains information about increasing-decreasing nature of $f\left(x\right)$ and $f\text{'}\left(x\right)$

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

• A

  (III) (iv) (P)
• B

  (I) (ii) (R)
• Correct answer
  C

  (II) (iii) (S)
• D

  (IV) (i) (S)

Hence, the correct answer is option: 'C'.