Applications of Derivatives
Roots and derivatives
Grade 12
Question:
<p>Let \(f'(x) \cdot g(x) - g'(x) \cdot f(x) = \left(x^2+1\right)^2 + 9\). If \(\alpha, \beta\) are consecutive roots of \(f(x) = 0\), which of the following are correct?</p>
<p>(a) \(g(\alpha) \cdot g(\beta) < 0\)</p>
<p>(b) \(f''(\alpha) g(\alpha) = 4\alpha(\alpha^2+1)\)</p>
<p>(c) \(\alpha\beta < 0\)</p>
<p>(d) There exists at least one root of \(f'(x)g(x) - g'(x)f(x) = 0\) in \((\alpha, \beta)\)</p>
Step-by-Step Solution
Key Concept: Recognize that f'(x)·g(x) - g'(x)·f(x) is the numerator of the quotient rule derivative d/dx[f(x)/g(x)]. Use the properties of consecutive roots and sign changes to determine behavior of f(x)/g(x) between roots.
<p><strong>Step 1:</strong> Recognize that f'(x)·g(x) - g'(x)·f(x) is the numerator of d/dx[f(x)/g(x)].</p><p>So: [d/dx(f/g)] × g² = (x²+1)² + 9</p><p><strong>Step 2:</strong> Since (x²+1)² + 9 ≥ 9 > 0 for all x ∈ ℝ, we have d/dx[f(x)/g(x)] > 0 everywhere (assuming g(x) ≠ 0).</p><p>This means f(x)/g(x) is strictly increasing on ℝ.</p><p><strong>Step 3:</strong> If α and β are consecutive roots of f(x) = 0 with α < β, then f(α) = 0 and f(β) = 0.</p><p>Between α and β, since f(x)/g(x) is strictly increasing and equals 0 at both endpoints, we must have g(x) → ±∞ at some point, or g(x) has opposite signs on (α, β).</p><p><strong>Step 4:</strong> Since f(x)/g(x) is strictly increasing and f(α) = f(β) = 0, there must be exactly one root of g(x) between α and β (where f/g changes from -∞ to +∞ or vice versa).</p><p><strong>Step 5:</strong> Between consecutive roots of f, there exists exactly one root of g(x). Both statements about the relationship between roots are correct.</p><p>∴ Answer: AC</p>
Correct Answer: AC