If <span class="math-inline">x^4 f\left( x \right) - \sqrt{1 - \sin 2\pi x} = |f\left( x \right)| - 2f\left( x \right)</span>, then f\left( -2 \right) equals
<span class="math-inline">\frac{1}{17}</span>
<span class="math-inline">\frac{1}{11}</span>
<span class="math-inline">\frac{1}{19}</span>
0
Step-by-Step Solution
Key Concept: Rearrange the equation to isolate f(x) terms, then analyze the constraint that √(1 - sin 2πx) must be real (requiring sin 2πx ≤ 1, which is always true) and determine when the equation is satisfied by considering the relationship between |f(x)| and f(x).
<p><strong>Step 1:</strong> Rearrange the given equation.</p><p>Given: x⁴f(x) - √(1 - sin 2πx) = |f(x)| - 2f(x)</p><p>Rearranging: x⁴f(x) + 2f(x) - |f(x)| = √(1 - sin 2πx)</p><p>This gives: f(x)(x⁴ + 2) - |f(x)| = √(1 - sin 2πx)</p></p><p><strong>Step 2:</strong> Evaluate at x = -2 (an integer).</p><p>When x = -2 (an integer), sin(2π(-2)) = sin(-4π) = 0</p><p>Therefore: √(1 - 0) = √1 = 1</p><p>The equation becomes: f(-2)(16 + 2) - |f(-2)| = 1</p><p>Simplifying: 18f(-2) - |f(-2)| = 1</p></p><p><strong>Step 3:</strong> Determine the sign of f(-2).</p><p>Case 1: If f(-2) ≥ 0, then |f(-2)| = f(-2)</p><p>So: 18f(-2) - f(-2) = 1</p><p>17f(-2) = 1</p><p>f(-2) = 1/17</p></p><p><strong>Step 4:</strong> Verify this solution is valid.</p><p>Since f(-2) = 1/17 > 0, our assumption that f(-2) ≥ 0 is correct.</p><p>Check: 18(1/17) - |1/17| = 18/17 - 1/17 = 17/17 = 1 ✓</p><p><strong>Step 5:</strong> Check the negative case.</p><p>If f(-2) < 0, then |f(-2)| = -f(-2)</p><p>So: 18f(-2) - (-f(-2)) = 1</p><p>19f(-2) = 1</p><p>f(-2) = 1/19</p><p>This contradicts f(-2) < 0, so this case is invalid.</p><p><strong>∴ Answer:</strong> A</p>
Correct Answer: A