Question

If \( f(x) = \left\{ \begin{array}{ll} mx + 1, & \text{when } x \leq \frac{\pi}{2} \\ \sin x + n, & \text{when } x > \frac{\pi}{2} \end{array} \right. \) is continuous at \( x = \frac{\pi}{2} \), then the values of \( m \) and \( n \) are:

• A. \( m = 1, n = 0 \)
• B. \( m = 0, n = 1 \)
• C. \( n = \frac{m\pi}{2} \)
• D. \( m = \frac{n\pi}{2} \)
• E. \( m = n = \frac{\pi}{2} \)

Hint:

For continuity at a point, the left-hand limit and the right-hand limit must be equal.

Answer 1

The Correct Option is C

For the function to be continuous at \( x = \frac{\pi}{2} \), we need to ensure that the left-hand limit and the right-hand limit at \( x = \frac{\pi}{2} \) are equal. 
From the left-hand side: \[ \lim_{x \to \frac{\pi}{2}^-} f(x) = m \left( \frac{\pi}{2} \right) + 1 \] From the right-hand side: \[ \lim_{x \to \frac{\pi}{2}^+} f(x) = \sin \left( \frac{\pi}{2} \right) + n = 1 + n \] Equating the two expressions for continuity: \[ m \left( \frac{\pi}{2} \right) + 1 = 1 + n \] \[ m \left( \frac{\pi}{2} \right) = n \] Thus, \( n = \frac{m\pi}{2} \).