by appropriately matching the information given in the three columns of the following…

by appropriately matching the information given in the three columns of the following table.
Let f(x) = x + log_ex – xlog_e x, x ∈ (0, ∞)
Column 1 contains information about zeros of f(x), f ′(x) and f ′′(x).
Column 2 contains information about the limiting behaviour 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 ∈ (1, e2)	(i)	(P) f is increasing in (0, 1)
(II)	f′(x) = 0 for some x ∈ (1, e)	(ii) \(\lim _{x \rightarrow \infty} f(x)=-\infty\)	(Q) f is decreasing in (e, e2)
(III)	f′(x) = 0 for some x ∈ (0, 1)	(iii)	(R) f′ is increasing in (0, 1)
(IV)	f′′(x) = 0 for some x ∈ (1, e)	(iv) \(\lim _{x \rightarrow \infty} f^{\prime \prime}(x)=0\)	(S) f′ is decreasing in (e, e2)

Which of the following options is the only incorrect combination?