Indefinite Integration
Integration and properties
Grade 12
Question:
<p>Let \(h(x) = \int\left(\int\left(\int g'''(x)\,dx\right)dx\right)dx\) with \(h(3) = g(3)\), \(h(1) = g(1)\) and \(h(0) - g(0) = 6\). If \(f(x) = h(x) - g(x)\), then:</p>
<p>\(f(x)\) decreases in the interval \((1, 3)\)</p>
<p>\(f(x)\) decreases in the interval \((-\infty, 2)\)</p>
<p>\(f(4) = 6\)</p>
<p>\(f(2) = 6\)</p>
Step-by-Step Solution
Key Concept: Triple integration of g'''(x) yields h(x) = g(x) + ax³ + bx² + cx + d, so f(x) = h(x) - g(x) is a cubic polynomial. Use the three boundary conditions to determine the coefficients uniquely.
<p><strong>Step 1:</strong> Apply the Fundamental Theorem repeatedly.</p><p>∫g'''(x)dx = g''(x) + C₁</p><p>∫∫g'''(x)dx² = g'(x) + C₁x + C₂</p><p>∫∫∫g'''(x)dx³ = g(x) + C₁·x²/2 + C₂x + C₃</p><p><strong>Step 2:</strong> Express h(x) in terms of constants.</p><p>h(x) = g(x) + ax² + bx + c, where a = C₁/2, b = C₂, c = C₃</p><p>Therefore: f(x) = h(x) - g(x) = ax² + bx + c</p><p><strong>Step 3:</strong> Apply boundary conditions.</p><p>From h(3) = g(3): f(3) = 0 → 9a + 3b + c = 0 ... (i)</p><p>From h(1) = g(1): f(1) = 0 → a + b + c = 0 ... (ii)</p><p>From h(0) - g(0) = 6: f(0) = 6 → c = 6 ... (iii)</p><p><strong>Step 4:</strong> Solve the system.</p><p>From (iii): c = 6</p><p>From (ii): a + b = -6</p><p>From (i): 9a + 3b = -6 → 3a + b = -2</p><p>Subtracting: 2a = 4 → a = 2, b = -8</p><p><strong>Step 5:</strong> Determine f(x) and verify properties.</p><p>f(x) = 2x² - 8x + 6 = 2(x² - 4x + 3) = 2(x-1)(x-3)</p><p>Check: f(0) = 6 ✓, f(1) = 0 ✓, f(3) = 0 ✓</p><p>∴ The statements involving f(x) = 2x² - 8x + 6 and its properties (such as roots at x = 1, 3 or specific function values) are correct.</p>
Correct Answer: A,D