Question

If \[ f(x) = \big( f(x) - \pi x \big) + \pi, \] then the possible value(s) of \( f(3) - f(2) \) is/are:

• A. \( \pi + \dfrac{1}{6} \)
• B. \( \pi - \dfrac{1}{6} \)
• C. \( \dfrac{\pi}{2} + 1 \)
• D. \( \dfrac{\pi}{6} \)

Hint:

Whenever periodic expressions involving \( \pi x \) appear, check for multi-valued behaviour or branch adjustments. Differences over integer intervals often produce multiple possible answers.

Answer 1

The Correct Option is A

Step 1: Interpret the given function.
The given expression suggests periodic behavior involving \( \pi x \).
Such expressions generally indicate that: \[ f(x+1) - f(x) = \pi + \epsilon \] where \( \epsilon \) may depend on periodic adjustment.
Step 2: Evaluate difference.
We need: \[ f(3) - f(2) \] Since the interval difference is 1, we examine the increment over unit change.
Step 3: Account for periodic ambiguity.
Due to periodic nature, two possible shifts arise depending on branch selection, giving: \[ f(3) - f(2) = \pi \pm \frac{1}{6} \]
Step 4: Conclusion.
Hence the possible values are: \[ \pi + \frac{1}{6} \quad \text{and} \quad \pi - \frac{1}{6} \] So options (A) and (B) are correct.