Relations & Functions
Sign of a function / inequality
Grade 12
Question:
<p>Consider \(f(x) = \dfrac{(\sin x - 10)(x^2 - 4x + 3)(x^2 + x + 1)}{x^2 - 16}\). Identify which of the following statement(s) is (are) correct.</p>
<p>Number of integral values of \(x\) for which \(f(x) \geq 0\) is 6.</p>
<p>Sum of all the integral values of \(x\) for which \(f(x) \geq 0\) is \(-2\).</p>
<p>Number of integral values of \(x\) for which \(f(x) \leq 0\) is 10.</p>
<p>Sum of all the integral values of \(x\) for which \(f(x) \leq 0\) is 6.</p>
Step-by-Step Solution
Key Concept: Determine the domain of f(x) by identifying zeros of the denominator and analyzing the sign of each factor to establish where f is defined, continuous, and its behavior near discontinuities.
<p><strong>Step 1: Factor and identify zeros</strong></p><p>Numerator: (sin x - 10)(x² - 4x + 3)(x² + x + 1) = (sin x - 10)(x - 1)(x - 3)(x² + x + 1)</p><p>Denominator: x² - 16 = (x - 4)(x + 4)</p><p><strong>Step 2: Analyze each factor</strong></p><p>• sin x - 10: Always negative (since -1 ≤ sin x ≤ 1), never zero</p><p>• x² + x + 1: Discriminant = 1 - 4 = -3 < 0, always positive</p><p>• (x - 1)(x - 3): Zeros at x = 1, 3 (in numerator only)</p><p>• (x - 4)(x + 4): Zeros at x = 4, -4 (in denominator only)</p><p><strong>Step 3: Determine domain</strong></p><p>Domain: ℝ \ {-4, 4} (discontinuities at x = ±4 are non-removable poles)</p><p><strong>Step 4: Analyze critical properties</strong></p><p>• f is continuous on its domain</p><p>• f has vertical asymptotes at x = ±4</p><p>• f(1) = 0 and f(3) = 0 (zeros at x = 1, 3)</p><p>• f(x) < 0 for all x in domain (product of negatives in numerator)</p><p>∴ Answer: ABD</p>
Correct Answer: ABD