Question

If $f(x) = \begin{cases} mx + 1, & x \le \frac{\pi}{2} \\ \sin x + n, & x > \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}, (m, n \in \mathbb{Z})$ then}

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

Hint:

Continuity at a point means you can simply "plug in" the value into both pieces of the function and set them equal.

Answer 1

The Correct Option is D

Step 1: Concept
For a function to be continuous at $x = a$, the Left Hand Limit (LHL), Right Hand Limit (RHL), and the value of the function at that point must be equal.

Step 2: Meaning
$\lim_{x \to (\pi/2)^-} f(x) = \lim_{x \to (\pi/2)^+} f(x) = f(\pi/2)$.

Step 3: Analysis
LHL $= m(\frac{\pi}{2}) + 1$. RHL $= \sin(\frac{\pi}{2}) + n = 1 + n$. Setting LHL $=$ RHL: $\frac{m\pi}{2} + 1 = 1 + n$. $\frac{m\pi}{2} = n$.

Step 4: Conclusion
The relation between the constants is $n = \frac{m\pi}{2}$. Final Answer: (D)