Applications of Derivatives
Discriminant Analysis
Grade 12
Question:
<p><strong>Ex. 18:</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 and distinct roots</p>
<p>(b) \(f(x)\) has three real roots but one of the roots would be repeated</p>
<p>(c) \(f(x)\) would have just one real root</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: When the discriminant of $f'(x)$ is zero, the cubic has a repeated critical point (inflection with horizontal tangent), typically resulting in one repeated root.
<p><strong>Step 1:</strong> If $D = 0$, then $f'(x) = 0$ has a repeated real root</p><p><strong>Step 2:</strong> The function $f(x)$ has an inflection point where the tangent is horizontal</p><p><strong>Step 3:</strong> At this point, $f(x)$ either touches the x-axis (giving a repeated root) or crosses it</p><p><strong>Step 4:</strong> In general, $f(x)$ has three real roots with at least one being repeated</p><p><strong>Step 5:</strong> The most common case is three real roots with one being a double root</p><p>∴ Answer is (b).</p>
Correct Answer: B