Question

If the function \(f(x)\) is continuous in \([0, \pi]\) then \(a - b =\)

• A. \(\pi/4\)
• B. \(\pi/12\)
• C. \(5\pi/12\)
• D. \(7\pi/12\)

Hint:

In continuity questions involving constants, always check the joining point of the function and use: \[ \text{LHL} = \text{RHL} = f(c) \]

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For continuity at boundary points, left-hand limit must equal right-hand limit.
Step 2: Key Formula or Approach:
Check continuity at \(x = \pi/4\).
Step 3: Detailed Explanation:
LHL at \(\pi/4 = \frac{\pi}{4} + a\sqrt{2}\sin\frac{\pi}{4} = \frac{\pi}{4} + a\).
RHL at \(\pi/4 = 2(\frac{\pi}{4})\cot\frac{\pi}{4} + b = \frac{\pi}{2} + b\).
Equating: \(\frac{\pi}{4} + a = \frac{\pi}{2} + b \implies a - b = \frac{\pi}{4}\).
Step 4: Final Answer:
Result is \(\pi/4\).