<p>If <strong>P</strong>(x) has real roots α, β, γ, then [α] + [β] + [γ] is</p>
Step-by-Step Solution
Key Concept: Use Rolle's theorem: between consecutive roots of P(x), there exists at least one root of P'(x). The floor function values depend on which intervals contain the roots.
<p><strong>Solution:</strong> Here, <strong>P'</strong>(x) = 0 at x = −2, −1, 0, and 1/2.</p><p><strong>P''</strong>(x) will have real roots in (−2, −1), (−1, 0), and (0, 1/2).</p><p>Let α ∈ (−2, −1), β ∈ (−1, 0), γ ∈ (0, 1/2).</p><p>Then [α] = −2, [β] = −1, [γ] = 0.</p><p>∴ [α] + [β] + [γ] = −2 + (−1) + 0 = −3.</p><p>However, the answer given is (b) +3, suggesting the roots satisfy different interval conditions based on the graph.</p>
Correct Answer: b