Step 1: Definition of acidic hydrogen
An acidic hydrogen is one that can be easily removed as a proton (\( \text{H}^+ \)), resulting in a stable conjugate base. The more stable the conjugate base, the more acidic the hydrogen.
Step 2: Analyze conjugate base stability
- In Option (B), the removal of the hydrogen leads to a carbanion which is resonance stabilized by conjugation with two adjacent double bonds. This forms an allylic anion extended over a conjugated cyclic system. - In Options (A), (C), and (D), although allylic positions are present, the resonance stabilization is less extensive compared to (B). Especially, (A) and (C) are sterically hindered or less effectively delocalized.
Step 3: Resonance in Option (B)
Removing the proton from the carbon adjacent to two double bonds gives a conjugate base that can delocalize the negative charge over a five-carbon conjugated system, similar to a cyclopentadienyl-like system — which is known for its aromatic stabilization when fully conjugated.
Conclusion: Option (B) has the most stable conjugate base, hence the most acidic hydrogen.
Final Answer: \( \boxed{\text{B}} \)
Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.
Let $ \mathbb{R} $ denote the set of all real numbers. Then the area of the region $$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x},\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0 \right\} $$ is