Applications of Derivatives
Rolle's theorem / roots of functions
Grade 12
Question:
<p>If \(y = f(x)\) be a concave upward function and \(y = g(x)\) be a function such that \(f'(x) \cdot g(x) - g'(x) \cdot f(x) = x^4 + 2x^2 + 10\), then</p>
<p>\(g(x)\) has at least one root between two consecutive roots of \(f(x) = 0\)</p>
<p>\(g(x)\) has at most one root between two consecutive roots of \(f(x) = 0\)</p>
<p>if α and β are two consecutive roots of \(f(x) = 0\), then \(\alpha\beta < 0\)</p>
<p>when \(f(x)\) increases \(g(x)\) decreases</p>
Step-by-Step Solution
Key Concept: Recognize that the given expression f'(x)·g(x) - g'(x)·f(x) is the numerator of the derivative of g(x)/f(x). Use this quotient rule structure combined with the concavity condition (f''(x) > 0) to deduce properties of the ratio.
<p><strong>Step 1:</strong> Recognize the given condition as a quotient derivative numerator.</p><p>The expression f'(x)·g(x) - g'(x)·f(x) is exactly the numerator in d/dx[g(x)/f(x)] = [f'g - g'f]/f²</p><p><strong>Step 2:</strong> Analyze the sign and properties.</p><p>Since f'(x)·g(x) - g'(x)·f(x) = x⁴ + 2x² + 10 > 0 for all x, we have d/dx[g(x)/f(x)] has the same sign as f(x).</p><p><strong>Step 3:</strong> Use concavity condition.</p><p>f''(x) > 0 (concave upward) means f is convex. Combined with f'g - g'f > 0, this constrains f(x) > 0 throughout (otherwise the quotient derivative would change sign inconsistently).</p><p><strong>Step 4:</strong> Conclude behavior of g/f.</p><p>Since f(x) > 0 and numerator is always positive, d/dx[g(x)/f(x)] > 0 always, so g(x)/f(x) is strictly increasing.</p><p>∴ Answer: AC</p>
Correct Answer: AC