Match List-I with List-II.

Choose the correct answer from the options given below:

(A)-(IV), (B)-(I), (C)-(III), (D)-(II)

The correct answer is Option (3) → (A)-(IV), (B)-(I), (C)-(III), (D)-(II)

(A) $f(x)=\frac{x}{\log_ex}$

$f'(x)=\frac{\log_ex-1}{(\log_ex)^2}$

for $f(x)$ to be increasing, $f'(x)>0$

$⇒\log_ex-1>0$

$⇒\log_ex>1$

$⇒x>e$

$⇒x∈(e, ∞)$ (IV)

(B) $f(x)=\frac{x}{2}+\frac{2}{x}=\frac{x^2+4}{2x}$

$f'(x)=\frac{(2x)(2x)-(x^2+4)(2)}{4x^2}$

$=\frac{4x^2-2x^2-8}{4x^2}=\frac{2x^2-8}{4x^2}$

$=\frac{x^2-4}{2x^2}$

$f'(x)>0$

$⇒x^2-4>0$

$⇒x>2,x<-2$

$⇒x∈(-∞,-2) (2,∞)$ (I)

(C) $f(x)=x^x$

$f'(x)=x^x+x^x.\log(x)$

$=x^x(1+\log (x))$

$⇒1+\log (x)>0$

$⇒\log (x)>-1$

$⇒x>\frac{1}{e}$

$⇒x∈\left(\frac{1}{e},∞\right)$ (III)

(D) $f(x)=\sin x-\cos t$

$f'(x)=\cos x$

$⇒\cos x>0$

$⇒x∈\left(-\frac{\pi}{4},\frac{\pi}{4}\right)$ (II)