Applications of Derivatives
Higher-order Derivatives and Critical Points
Grade 12
Question:
<p>The minimum number of real roots of equation <span style="text-decoration:overline">P''</span>(x))² + <strong>P'</strong>(x) · <strong>P'''</strong>(x) = 0 is</p>
<p>(a) 5</p>
<p>(b) 7</p>
<p>(c) 6</p>
<p>(d) 4</p>
Step-by-Step Solution
Key Concept: Recognize that the equation represents critical points of the product <strong>P'</strong>(x)·<strong>P''</strong>(x). Use the product rule and properties of derivatives to count roots.
<p><strong>Solution:</strong> The equation can be written as:</p><p>$$(<strong>P''</strong>(x))^2 + <strong>P'</strong>(x) \cdot <strong>P'''</strong>(x) = 0$$</p><p>This is equivalent to $$\frac{d}{dx}[<strong>P'</strong>(x) \cdot <strong>P''</strong>(x)] = 0$$</p><p>Let g(x) = <strong>P'</strong>(x) · <strong>P''</strong>(x). We need to find zeros of g'(x).</p><p>From the graph, <strong>P'</strong>(x) has 4 real roots and <strong>P''</strong>(x) has 3 real roots.</p><p>By analyzing the critical points and behavior, the minimum number of roots is 6.</p>
Correct Answer: c