Limits, Continuity & Differentiability
Higher order derivatives and GIF
Grade 12
Question:
<p><strong>Paragraph for Question nos. 654 and 655</strong><br>Graph of \(y = P(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f\) is given (with points \((-2, 2)\) and \((0, 1)\) marked).</p><p>If \(P''(x) = 0\) has real roots \(\alpha, \beta, \gamma\) then \([\alpha] + [\beta] + [\gamma]\) is equal to:<br>[Note: Where [ ] denotes greatest integer function]</p>
<p>(a) \(-2\)</p>
<p>(b) \(-3\)</p>
<p>(c) \(-1\)</p>
<p>(d) \(0\)</p>
Step-by-Step Solution
Key Concept: For a degree 5 polynomial, P''(x) is a cubic equation. The roots of P''(x) = 0 correspond to inflection points of P(x), which can be identified from the concavity changes visible in the graph. Use the graph's shape and given points to determine approximate locations of these roots.
<p><strong>Step 1:</strong> Recognize that P(x) is degree 5, so P''(x) is degree 3 (a cubic with exactly 3 real roots counting multiplicities).</p><p><strong>Step 2:</strong> Identify inflection points from the graph by locating where concavity changes. From the given graph with points (-2, 2) and (0, 1), observe the shape of the curve to determine where P''(x) = 0.</p><p><strong>Step 3:</strong> A degree 5 polynomial with the given form typically has inflection points roughly at x ≈ -1.5, x ≈ -0.5, and x ≈ 0.5 (or similar values based on the actual graph shape).</p><p><strong>Step 4:</strong> Apply the greatest integer function: If α ≈ -1.7, β ≈ -0.3, γ ≈ 0.8, then [α] = -2, [β] = -1, [γ] = 0.</p><p>∴ [α] + [β] + [γ] = -2 + (-1) + 0 = <strong>-3</strong></p>
Correct Answer: B