Question

Let $ 𝑓 \left(x\right) = 𝑥 + log_{e} x − 𝑥 log_{e} 𝑥, 𝑥 ∈ \left(0,∞\right).$
• Column 1 contains information about zeros of $𝑓 (𝑥) , 𝑓'(𝑥)$ and $f ''(𝑥)$.
• Column 2 contains information about the limiting behavior 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 $x\in\left(1,e^{2}\right)$	(i) $\lim_{x\to\infty}f \left(x\right)=0$	(P) $𝑓$ is increasing in ($0, 1$)
(II) $𝑓(𝑥) = 0$ for some $x\in\left(1, e\right)$	(ii) $\lim_{x\to\infty}f \left(x\right)=-\infty$	(Q) $f$ is decreasing in ($𝑒, 𝑒^2$)
(III) $𝑓(𝑥) = 0$ for some $x\in\left(0, 1\right)$	(ii) $\lim_{x\to\infty}f' \left(x\right)=-\infty$	(R) $f'$ is increasing in ($0, 1$)
(IV) $f ''(𝑥) = 0$ for some $x\in\left(1, e\right)$	(ii) $\lim_{x\to\infty}f'' \left(x\right)=0$	(S) $f'$ is decreasing in ($𝑒, 𝑒^2$)

Which of the following options is the only INCORRECT combination?

• A. (I) (iii) (P)
• B. (II) (iv) (Q)
• C. (III) (i) (R)
• D. (II) (iii) (P)

Answer 1

Answer: (C)

1) $f ( x )= x +\log _{e}( x )- x \log _{e}( x )$
2) $f ^{\prime}( x )=\frac{1}{ x }-\log _{ e }( x )$
3) $f^{\prime \prime}(x)=-\frac{(x+1)}{x^{2}}<0 \forall x>0$
4) $f (1)= f ( e )=1, f \left( e ^{2}\right)<0$
5) $f ^{\prime}(1)=1, f ^{\prime}( e )=\frac{1}{ e }-1<0$