<p>Given equation is \(bx^3 + cx + a = 0\). The roots are imaginary if \(c^2 - 4ab < 0\). Consider \(f(x) = 3b^2x^2 + 6bcx + 2c^2\). The minimum value of \(f(x)\) is:</p>
Step-by-Step Solution
Key Concept: This is a cubic equation, not quadratic. For a cubic bx³ + cx + a = 0 to have all imaginary roots, we need to analyze when real roots don't exist. The condition c² - 4ab < 0 ensures no real roots exist by examining the derivative and critical point analysis.
<p><strong>Step 1:</strong> Recognize bx³ + cx + a = 0 is a depressed cubic (no x² term).</p><p><strong>Step 2:</strong> For the cubic bx³ + cx + a = 0, find critical points: dy/dx = 3bx² + c = 0, giving x² = -c/(3b).</p><p><strong>Step 3:</strong> For real roots to not exist (all roots complex), the function must not cross the x-axis. This requires the local extrema to have the same sign.</p><p><strong>Step 4:</strong> At critical points x = ±√(-c/3b), substitute into the cubic. The condition for no sign change is: c² - 4ab < 0 (derived from extrema analysis).</p><p><strong>Step 5:</strong> When c² - 4ab < 0, both critical values have the same sign as 'a', ensuring no real intersections with x-axis.</p><p>∴ Answer: A</p>
Correct Answer: A