<p>The figure shows \(y = P(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f\). If \(P''(x)\) has real roots \(\alpha, \beta, \gamma\), then find \([\alpha] + [\beta] + [\gamma]\), where \([\cdot]\) denotes the greatest integer function.</p>
Step-by-Step Solution
Key Concept: Use Rolle's theorem to determine the intervals containing roots of the second derivative based on the roots of the first derivative visible in the graph.
<p><strong>Solution:</strong> From the figure, $P'(x) = 0$ at $x = -2, -1, 0, \frac{1}{2}$.</p><p>By Rolle's theorem, between any two roots of $P'(x)$, there exists at least one root of $P''(x)$.</p><p>Therefore, $P''(x)$ has real roots in the intervals $(-2, -1)$, $(-1, 0)$, and $(0, \frac{1}{2})$.</p><p>Let $\alpha \in (-2, -1)$, $\beta \in (-1, 0)$, $\gamma \in (0, \frac{1}{2})$.</p><p>Then: $[\alpha] = -2$, $[\beta] = -1$, $[\gamma] = 0$</p><p>$$[\alpha] + [\beta] + [\gamma] = -2 + (-1) + 0 = -3$$</p><p>∴ Answer is (b).</p>
Correct Answer: B