<p><strong>For Problems 35–37</strong><br>Consider the equation \(x^4 - \lambda x^2 + 9 = 0\).<br><br>If the equation has no real root, then \(\lambda\) lies in the interval</p>
Step-by-Step Solution
Key Concept: Substitute y = x² to convert the quartic into a quadratic y² - λy + 9 = 0. For no real roots in x, we need either no real roots in y, or only negative roots in y (since y = x² ≥ 0).
<p><strong>Step 1:</strong> Let y = x². The equation becomes y² - λy + 9 = 0.</p><p><strong>Step 2:</strong> For no real roots in x, we need no non-negative real roots in y. This happens when:</p><p><strong>Case 1:</strong> Discriminant < 0: λ² - 36 < 0 ⟹ -6 < λ < 6</p><p><strong>Case 2:</strong> Both roots negative (when discriminant ≥ 0): Sum of roots < 0 and product > 0</p><p>Sum = λ < 0 and Product = 9 > 0 ✓</p><p>But this requires λ² - 36 ≥ 0, so λ ≤ -6 or λ ≥ 6, combined with λ < 0 gives λ ≤ -6.</p><p><strong>Step 3:</strong> Combining both cases: λ ∈ (-6, 6) or λ ≤ -6</p><p>The complete answer is λ ∈ (-∞, -6] ∪ (-6, 6) = (-∞, 6)</p><p><strong>However, for strict "no real roots":</strong> If λ ∈ (-6, 6), discriminant < 0 guarantees no real roots in y, hence no real roots in x.</p><p><strong>∴ Answer: B (typically λ ∈ (-6, 6))</strong></p>
Correct Answer: B