Let 𝑓 (x) = 𝑥 + loge x − 𝑥 loge 𝑥, 𝑥 ∈ (0,∞). • 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∈(1,e2) (i) limx→∞f (x)=0 (P) 𝑓 is increasing in (0, 1) (II) 𝑓(𝑥) = 0 for some x∈(1, e) (ii) limx→∞f (x)=-∞ (Q) f is decreasing in (𝑒, 𝑒2) (III) 𝑓(𝑥) = 0 for some x∈(0, 1) (ii) limx→∞f' (x)=-∞ (R) f' is increasing in (0, 1) (IV) f ''(𝑥) = 0 for some x∈(1, e) (ii) limx→∞f'' (x)=0 (S) f' is decreasing in (𝑒, 𝑒2) Which of the following options is the only CORRECT combination?

Question

Let 𝑓 (x) = 𝑥 + loge x − 𝑥 loge 𝑥, 𝑥 ∈ (0,∞). • 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∈(1,e2) (i) limx→∞f (x)=0 (P) 𝑓 is increasing in (0, 1) (II) 𝑓(𝑥) = 0 for some x∈(1, e) (ii) limx→∞f (x)=-∞ (Q) f is decreasing in (𝑒, 𝑒2) (III) 𝑓(𝑥) = 0 for some x∈(0, 1) (ii) limx→∞f' (x)=-∞ (R) f' is increasing in (0, 1) (IV) f ''(𝑥) = 0 for some x∈(1, e) (ii) limx→∞f'' (x)=0 (S) f' is decreasing in (𝑒, 𝑒2) Which of the following options is the only CORRECT combination? (A) (I) (ii) (R) (B) (II) (iii) (S) (C) (III) (iv) (P) (D) (IV) (i) (S)

Tardigrade Admin · Accepted Answer

1) f ( x )= x + log e( x )- x log e( x ) 2) f prime( x )=(1/ x )- log e ( x ) 3) f prime prime(x)=-((x+1)/x2)<0 ∀ x>0 4) f (1)= f ( e )=1, f ( e 2)<0 5) f prime(1)=1, f prime( e )=(1/ e )-1<0

Column 1	Column 2	Column 3
(I) $f (x) = 0$ for some $x \in (1, e^{2})$	(i) $lim_{x \to \infty} f (x) = 0$	(P) $f$ is increasing in ( $0, 1$ )
(II) $f (x) = 0$ for some $x \in (1, e)$	(ii) $lim_{x \to \infty} f (x) = - \infty$	(Q) $f$ is decreasing in ( $e, e^{2}$ )
(III) $f (x) = 0$ for some $x \in (0, 1)$	(ii) $lim_{x \to \infty} f^{'} (x) = - \infty$	(R) $f^{'}$ is increasing in ( $0, 1$ )
(IV) $f^{''} (x) = 0$ for some $x \in (1, e)$	(ii) $lim_{x \to \infty} f^{''} (x) = 0$	(S) $f^{'}$ is decreasing in ( $e, e^{2}$ )

(I) (ii) (R)

(II) (iii) (S)

(III) (iv) (P)

(IV) (i) (S)

1) f(x)=x+loge​(x)−xloge​(x) 2) f′(x)=x1​−loge​(x) 3) f′′(x)=−x2(x+1)​<0∀x>0 4) f(1)=f(e)=1,f(e2)<0 5) f′(1)=1,f′(e)=e1​−1<0

1) $f (x) = x + lo g_{e} (x) - x lo g_{e} (x)$
2) $f^{'} (x) = \frac{1}{x} - lo g_{e} (x)$
3) $f^{''} (x) = - \frac{( x + 1 )}{x ^{2}} < 0\forall x > 0$
4) $f (1) = f (e) = 1, f (e^{2}) < 0$
5) $f^{'} (1) = 1, f^{'} (e) = \frac{1}{e} - 1 < 0$