Which of the following options is the only CORRECT combination?

In column 1
f(1)=1+0−(1)(0)=1
f(e^2)=e^2+2−(2)e^2=2−e^2<0
Since f(x) is a continuous function, f(x) will be zero for some x where 1<x<e^2
Therefore (I) is correct
f′(x)=1/x−lnx
f′(1)=1
f′(e)=1/e−1<0
Since f′(x) is a continuous function, f′(x) will be zero for some x where 1<x<e
Therefore (II) is correct
f′(0)→∞−(−∞)→∞+∞>0
f′(1)=1
Since f′(x) is a continuous function, f′(x) will not be zero for some x where 0<x<1
Therefore (III) is false
f″(x)=−1/x^2−1/x
Therefore f″(x)<0 for x>1
Therefore (IV) is false
Therefore option C, D can be eliminated
In column 3
f′(x)>0 for 0<x<1
Therefore f(x) will be increasing in (0,1)
Therefore (P) is correct
f′(x)<0 for e<x<e^2
Therefore f(x) will be decreasing in (e,e^2)
Therefore (Q) is correct
f″(x)=−1/x^2−1/x
f″(x)<0 for x>0
Therefore f′(x) is decreasing for x>0
Therefore S is correct and R is false
Therefore option B can be eliminated
In column 2
lim_{x→∞}f′(x)=lim_{x→∞}(1/x−lnx)=0−∞=−∞
Therefore (iii) is true
Therefore answer is option A