<p><strong>Example 54:</strong> If the function <span>f(x) = x^3 + 9x^2 - 24x + c</span> has three real and distinct roots <span>N, P</span> and <span>S</span>, then the value of <span>[N] + [P] + [S]</span> are</p>
Step-by-Step Solution
Key Concept: Use calculus to find critical points and determine when the function has three real roots. The parameter c determines the intervals where roots lie.
<p><strong>Step 1:</strong> Let <span>y = x^3 + 9x^2 - 24x + c</span></p><p><strong>Step 2:</strong> Find <span>\frac{dy}{dx} = 3x^2 + 18x - 24 = 3(x^2 + 6x - 8) = 3(x + 2)(x + 4)</span></p><p><strong>Step 3:</strong> Critical points are at <span>x = -2</span> and <span>x = -4</span></p><p><strong>Step 4:</strong> For three real and distinct roots, we need <span>f(-2) \cdot f(-4) < 0</span></p><p><strong>Step 5:</strong> This gives <span>(c - 20)(c - 16) < 0</span>, so <span>c \in (-20, -16)</span></p><p><strong>Step 6:</strong> When <span>c \in (-20, -18)</span>: <span>N \in (1, 2), P \in (2, 3), S \in (4, 5)</span>, so <span>[N] + [P] + [S] = 1 + 2 + 4 = 7</span></p><p><strong>Step 7:</strong> When <span>c \in (-18, -16)</span>: <span>N \in (1, 2), P \in (3, 4), S \in (4, 5)</span>, so <span>[N] + [P] + [S] = 1 + 3 + 4 = 8</span></p><p>∴ Answer is (c) 7, 8.</p>
Correct Answer: C