Given function

$f(x)=x+\ln x-x\ln x, \qquad x\in(0,\infty)$

Find derivatives

$f'(x)=1+\frac1x-(\ln x+1)=\frac1x-\ln x$

$f''(x)=-\frac1{x^2}-\frac1x=-\frac{1+x}{x^2}<0 \quad \forall x>0$

Match Column 1

We analyze the zeroes.

(I) Zero of $f(x)$

Solve approximately: $x+\ln x-x\ln x=0$

Check values: $f(1)=1>0$ $f(e^2)=e^2+2-2e^2=2-e^2<0$

Since $f$ is continuous, there is some root in $(1,e^2)$.

So (I) is true.

Zero of $f'(x)$

$f'(x)=\frac1x-\ln x$

Check: $f'(1)=1>0, \qquad f'(e)=\frac1e-1<0$

Hence by continuity, $f'(x)=0$ for some $x\in(1,e)$.

Also for $x\in(0,1)$, $\ln x<0, \quad \frac1x>0 \implies f'(x)=\frac1x-\ln x>0$ so there is no zero in $(0,1)$.

Therefore:

• statement (ii): true
• statement (iii): false
• statement (iv) same as (ii): true

Match Column 2

Limit of $f(x)$ as $x\to\infty$

$f(x)=x+\ln x-x\ln x=x(1-\ln x)+\ln x$

Since $x\ln x$ dominates $x$ and $\ln x$, $\lim_{x\to\infty} f(x)=-\infty$

So (ii) is correct, (i) is false.

Limit of $f'(x)$ as $x\to\infty$

$f'(x)=\frac1x-\ln x \to 0-\infty=-\infty$

So (iii) is correct, (iv) is false.

Match Column 3

Since $f'(x)=\frac1x-\ln x$

• On $(0,1)$, $\ln x<0$, so $f'(x)>0$; hence $f$ is increasing on $(0,1)$.
• On $(e,e^2)$, $\ln x>1$ and $1/x<1/e<1$, so $f'(x)<0$; hence $f$ is decreasing on $(e,e^2)$.

Also, $f''(x)=-\frac{1+x}{x^2}<0$ for all $x>0$, hence $f'$ is decreasing on every interval, including $(e,e^2)$. Therefore:

• P: $f$ increasing in $(0,1)$ → true
• Q: $f$ decreasing in $(e,e^2)$ → true
• R: $f'$ increasing in $(0,1)$ → false
• S: $f'$ decreasing in $(e,e^2)$ → true

From the options, the third-column labels used are:

• Option A uses (P) = $f$ increasing in $(0,1)$
• Option B uses (Q) = $f$ decreasing in $(e,e^2)$
• Option C uses (P) = $f$ increasing in $(0,1)$
• Option D uses (R) = $f'$ increasing in $(0,1)$

Check each option

Option A: $(I),(iii),(P)$

• $(I)$ : true
• $(iii)$ in Column 2 means $\lim_{x\to\infty} f'(x)=-\infty$ : true
• $(P)$ : true

So A is correct combination.

Option B: $(II),(iv),(Q)$

• $(II)$ : $f'(x)=0$ for some $x\in(1,e)$ : true
• $(iv)$ in Column 2: $\lim_{x\to\infty} f'(x)=0$ : false
• $(Q)$ : true

So B is incorrect combination.

Option C: $(II),(ii),(P)$

• $(II)$ : true
• $(ii)$ in Column 2: $\lim_{x\to\infty} f(x)=-\infty$ : true
• $(P)$ : true

So C is correct combination.

Option D: $(III),(i),(R)$

• $(III)$ : $f'(x)=0$ for some $x\in(0,1)$ : false
• $(i)$ in Column 2: $\lim_{x\to\infty} f(x)=0$ : false
• $(R)$ : false

So D is also incorrect combination.

Conclusion

From direct analysis, both B and D are incorrect, so the statement "only incorrect combination" is inconsistent with the data as typed.

However, among the given single-choice options, D is certainly incorrect, and this matches the stored answer.

But strictly speaking, B is also incorrect.