Applications of Derivatives
Critical Points and Roots
Grade 12
Question:
<p><strong>Ex. 17:</strong> If \(D = 4(a^2 - 3b) > 0\) and \(f(x_1) \cdot f(x_2) = 0\), where \(x_1, x_2\) are the roots of \(f'(x) = 0\), 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) \(f(x)\) has three real roots but all are same</p>
Step-by-Step Solution
Key Concept: When a local extremum of $f(x)$ lies on the x-axis, that point is a repeated root of $f(x)$.
<p><strong>Step 1:</strong> If $D > 0$, then $f'(x) = 0$ has two distinct real roots $x_1$ and $x_2$</p><p><strong>Step 2:</strong> The condition $f(x_1) \cdot f(x_2) = 0$ means at least one of $f(x_1)$ or $f(x_2)$ is zero</p><p><strong>Step 3:</strong> This means the curve touches the x-axis at a local extremum point</p><p><strong>Step 4:</strong> Therefore, one root is repeated (the curve is tangent to the x-axis at that point)</p><p><strong>Step 5:</strong> The function has three real roots with one being repeated</p><p>∴ Answer is (b).</p>
Correct Answer: B