<p>If <span>\( y = \dfrac{x^2 - x + c}{x^2 + x + c} \)</span> and <span>\( y \leq 1 - \dfrac{2}{1 - 2\sqrt{c}} = \dfrac{5}{3} \)</span>, find the value of <span>\(c\)</span>.</p>
Step-by-Step Solution
Key Concept: Rearrange the given inequality by substituting the right-hand side value, then manipulate the rational function algebraically to extract constraints on c. The bound on y must hold for all valid x, which means analyzing the range of the rational function.
<p><strong>Step 1:</strong> From the given condition, first find c using: 1 - 2/(1-2√c) = 5/3</p><p>⟹ -2/(1-2√c) = 5/3 - 1 = 2/3</p><p>⟹ -2/(1-2√c) = 2/3</p><p>⟹ 1 - 2√c = -3</p><p>⟹ 2√c = 4</p><p>⟹ √c = 2</p><p>⟹ <strong>c = 4</strong></p><p><strong>Step 2:</strong> Verify by checking the range condition. With c = 4, we have y = (x² - x + 4)/(x² + x + 4). For y ≤ 5/3 to hold:</p><p>3(x² - x + 4) ≤ 5(x² + x + 4)</p><p>3x² - 3x + 12 ≤ 5x² + 5x + 20</p><p>0 ≤ 2x² + 8x + 8 = 2(x² + 4x + 4) = 2(x+2)²</p><p>This is always true, confirming the constraint is satisfied.</p><p>∴ <strong>Answer: c = 4</strong></p>
Correct Answer: 4