Applications of Derivatives
Extrema and Roots
Grade 12
Question:
<p><strong>Ex. 15:</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) None of the above</p>
Step-by-Step Solution
Key Concept: When $f'(x) = 0$ has two real roots creating a local extremum with opposite signs, the cubic crosses the x-axis three times, giving three distinct real roots.
<p><strong>Step 1:</strong> If $D > 0$, then $f'(x) = 0$ has two distinct real roots $x_1 < x_2$</p><p><strong>Step 2:</strong> The function $f(x)$ has a local maximum at $x_1$ and a local minimum at $x_2$</p><p><strong>Step 3:</strong> The condition $f(x_1) \cdot f(x_2) < 0$ means $f(x_1)$ and $f(x_2)$ have opposite signs</p><p><strong>Step 4:</strong> If the local maximum is positive and the local minimum is negative (or vice versa), the curve crosses the x-axis three times</p><p><strong>Step 5:</strong> Therefore, $f(x)$ has three distinct real roots</p><p>∴ Answer is (a).</p>
Correct Answer: A