Applications of Derivatives
Monotonicity and Roots
Grade 12
Question:
<p><strong>Ex. 14:</strong> If \(D = 4(a^2 - 3b) < 0\), then \(f(x) = x^3 + ax^2 + bx + c\)</p>
<p>(a) \(f(x)\) has all real roots</p>
<p>(b) \(f(x)\) has one real and two imaginary roots</p>
<p>(c) \(f(x)\) has repeated roots</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: The sign of the discriminant of $f'(x)$ determines the number of critical points and the monotonicity of $f(x)$, which in turn determines the number of real roots.
<p><strong>Step 1:</strong> The discriminant of $f'(x) = 3x^2 + 2ax + b$ is $D = 4a^2 - 12b = 4(a^2 - 3b)$</p><p><strong>Step 2:</strong> If $D < 0$, then $f'(x)$ has no real roots, so $f'(x) > 0$ for all $x \in \mathbb{R}$</p><p><strong>Step 3:</strong> This means $f(x)$ is strictly monotonically increasing on $\mathbb{R}$</p><p><strong>Step 4:</strong> A strictly monotonic cubic function crosses the x-axis exactly once</p><p><strong>Step 5:</strong> Therefore, $f(x)$ has one real root and two complex conjugate imaginary roots</p><p>∴ Answer is (b).</p>
Correct Answer: B