Question

Match List-I with List-II.
\[ \begin{array}{|c|c|} \hline \textbf{List-I (Function)} & \textbf{List-II (Interval in which function is increasing)} \\ \hline \frac{x}{\log_e x} & (-\infty, -2) \cup (2, \infty) \\ \hline \frac{x}{2} + \frac{2}{x}, x \neq 0 & \left(-\frac{\pi}{4}, \frac{\pi}{4}\right) \\ \hline x^x, x > 0 & \left(\frac{1}{e}, \infty\right) \\ \hline \sin x - \cos x & (e, \infty) \\ \hline \end{array} \]
Choose the correct answer from the options given below:

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

Answer 1

The Correct Option is C

To resolve this matching problem, we analyze each function from List-I by determining its intervals of increase. The analysis proceeds as follows:

(A) \(\frac{x}{\log_e x}\)

The derivative of \(\frac{x}{\log_e x}\) is calculated as \( f'(x) = \frac{\log_e x - 1}{(\log_e x)^2} \). For the function to be increasing, \(f'(x)>0\), which implies \(\log_e x - 1>0\). This inequality simplifies to \(\log_e x>1\), yielding \(x>e\). Thus, the function increases on the interval \((e, \infty)\).

(B) \(\frac{x}{2} + \frac{2}{x}, x eq 0\)

The derivative is \( f'(x) = \frac{1}{2} - \frac{2}{x^2} \). Setting \(f'(x)>0\) leads to \(\frac{1}{2}>\frac{2}{x^2}\), which simplifies to \(x^2>4\). The solutions are \(x>2\) or \(x<-2\). Therefore, the function is increasing on \((-\infty, -2) \cup (2, \infty)\).

(C) \(x^x, x > 0\)

Using logarithmic differentiation, we find the derivative of \(f(x) = x^x\) to be \(f'(x) = x^x (\log x + 1)\). For \(f'(x)>0\), we require \(\log x + 1>0\), which means \(\log x>-1\). This implies \(x>\frac{1}{e}\). The function increases on the interval \(\left(\frac{1}{e}, \infty\right)\).

(D) \(\sin x - \cos x\)

The derivative is \( f'(x) = \cos x + \sin x \). To find where \(f'(x)>0\), we use the identity \(\cos x + \sin x = \sqrt{2} \sin(x + \frac{\pi}{4})\). The condition becomes \(\sqrt{2} \sin(x + \frac{\pi}{4})>0\), or \(\sin(x + \frac{\pi}{4})>0\). This holds when \(0<x + \frac{\pi}{4}<\pi\), which simplifies to \(-\frac{\pi}{4}<x<\frac{3\pi}{4}\). The provided range in List-II is \((- \frac{\pi}{4}, \frac{\pi}{4})\), which is a sub-interval where the function is increasing.

Matching Results:

The identified intervals of increase for each function in List-I are compared with the options in List-II:

List-I (Function)	List-II (Interval in which function is increasing)
(A) \(\frac{x}{\log_e x}\)	(e, \infty)
(B) \(\frac{x}{2} + \frac{2}{x}, x eq 0\)	(-\infty, -2) \cup (2, \infty)
(C) \(x^x, x > 0\)	\(\left(\frac{1}{e}, \infty\right)\)
(D) \(\sin x - \cos x\)	\((- \frac{\pi}{4}, \frac{\pi}{4})\)

The correct matching is: (A-IV), (B-I), (C-III), (D-II)