Which of the following options is the only INCORRECT combination?

In column 1
f(1)=1+0−(1)(0)=1
f(e²)=e²+2−(2)e²=2−e²<0
Since f(x) is a continuous function, f(x) will be zero for some x where 1<x<e²
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/x²−1/x
Therefore f′′(x)<0 for x>1
Therefore (IV) is false
Since we need to find the incorrect option, only option D has (III), so option D will be our answer