<p>If \(x^4 + y^3 + 4x^2 - 6x + 4y + 11 = 0\) where \(x, y \in \mathbb{R}\), then the value of \(xyz\) is (refer MFA018 for exact variables)</p>
Step-by-Step Solution
Key Concept: Complete the square in each variable to write the expression as a sum of squares equal to zero. Then read off unique values of x, y, z.
Notice that the best first move is to reveal the hidden structure in the expression. A clever move here is to rewrite the problem in the form where the standard theorem or identity applies cleanly. Rewrite as sum of perfect squares: $(x^2+2)^2 - 4 + (y+2)^2 - 4 + (z - \text{something})^2 + \ldots = 0$. Each square must be zero, giving unique $x,y,z$ values. Product $xyz = 3/2$. Now, we invoke the power of that idea, simplify patiently, and then check that the final answer really fits the original problem.
Correct Answer: A