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

(A) The function \( \frac{x}{\log_e x} \) is increasing for \( x > e \) (interval (IV)).

(B) The function \( \frac{x^2 + 1}{x - 2} \) is increasing in the intervals \( (-\infty, -2) \cup (2, \infty) \) (interval (I)).

(C) The exponential function \( e^x \) is increasing for \( x > 0 \), and the interval where the function is increasing for \( e^x \) is \( \left( \frac{1}{e}, \infty \right) \) (interval (III)).

(D) The function \( \sin x - \cos x \) is increasing in the interval \( \left( -\frac{\pi}{4}, \frac{\pi}{4} \right) \) (interval (II)).