<p>Let \(\int \phi(x) \sin x\, dx = \cos(-\phi(x) + \phi''(x) - \phi''''(x) + \ldots) + \sin x(\phi'(x) - \phi'''(x) + \phi'''''(x) - \ldots)\), where \(g(x) = -x^4 - 2x^2 - 1 + 12x^2 + 4 - 24 = -x^4 + 10x^2 - 21\) and \(f(x) = 4x^3 + 4x - 24x = 4x^3 - 20x\).</p><p>Match the following:</p><p>(P) \(f(x)\) is many-one onto function.</p><p>(Q) \(g(x)\) is many-one into function.</p><p>(R) Number of points where \(|f(x)|\) is non-derivable is 3.</p><p>(S) Number of points where \(|g(x)|\) is non-derivable is 4.</p>
Step-by-Step Solution
Key Concept: To determine if functions are one-one/onto and find non-derivability points of absolute value functions, analyze critical points using derivatives and examine where the original functions change sign.
<p><strong>Step 1: Analyze f(x) = 4x³ - 20x</strong></p><p>f'(x) = 12x² - 20</p><p>f'(x) = 0 ⟹ x² = 5/3 ⟹ x = ±√(5/3)</p><p>Since f'(x) is quadratic with positive leading coefficient, f'(x) < 0 for x ∈ (-√(5/3), √(5/3)) and f'(x) > 0 otherwise.</p><p>f(x) has a local maximum and local minimum, so it is many-one (not one-one). As x → ±∞, f(x) → ±∞, so range = ℝ (onto).</p><p><strong>(P) is TRUE</strong>: f(x) is many-one onto.</p><p><strong>Step 2: Analyze g(x) = -x⁴ + 10x² - 21</strong></p><p>g'(x) = -4x³ + 20x = -4x(x² - 5)</p><p>g'(x) = 0 ⟹ x = 0, ±√5</p><p>Critical points at x = 0 (local min), x = ±√5 (local max).</p><p>g(√5) = -(25) + 10(5) - 21 = -25 + 50 - 21 = 4</p><p>g(0) = -21</p><p>As x → ±∞, g(x) → -∞</p><p>Range of g(x) is (-∞, 4], which is a proper subset of ℝ.</p><p><strong>(Q) is TRUE</strong>: g(x) is many-one into (multiple x-values map to same output, range ⊂ ℝ).</p><p><strong>Step 3: Find non-derivability points of |f(x)| = |4x³ - 20x|</strong></p><p>|f(x)| is non-derivable where f(x) = 0.</p><p>f(x) = 0 ⟹ 4x(x² - 5) = 0 ⟹ x = 0, ±√5</p><p>Three solutions: x = -√5, 0, √5</p><p><strong>(R) is TRUE</strong>: Number of non-derivable points of |f(x)| is 3.</p><p><strong>Step 4: Find non-derivability points of |g(x)| = |-x⁴ + 10x² - 21|</strong></p><p>|g(x)| is non-derivable where g(x) = 0.</p><p>-x⁴ + 10x² - 21 = 0 ⟹ x⁴ - 10x² + 21 = 0</p><p>Let u = x²: u² - 10u + 21 = 0</p><p>(u - 3)(u - 7) = 0 ⟹ u = 3 or u = 7</p><p>x² = 3 ⟹ x = ±√3</p><p>x² = 7 ⟹ x = ±√7</p><p>Four solutions: x = -√7, -√3, √3, √7</p><p><strong>(S) is TRUE</strong>: Number of non-derivable points of |g(x)| is 4.</p><p><strong>∴ Answer:</strong> (P)→True, (Q)→True, (R)→3, (S)→4</p>
Correct Answer: (P)→True, (Q)→True, (R)→3, (S)→4