<p>Let <br/> \( f(x) = \dfrac{3x^2 + 9x + 17}{3x^2 + 9x + 7} \) <br/> Find the maximum value of \( f(x) \).</p>
Step-by-Step Solution
Key Concept: Substitute u = 3x² + 9x + 7 to transform f(x) into a rational function of u, then find the range by analyzing how the numerator and denominator relate through u.
<p><strong>Step 1:</strong> Recognize the relationship between numerator and denominator.</p><p>Numerator = 3x² + 9x + 17 = (3x² + 9x + 7) + 10</p><p><strong>Step 2:</strong> Let u = 3x² + 9x + 7. Then:</p><p>f(x) = (u + 10)/u = 1 + 10/u</p><p><strong>Step 3:</strong> Find the range of u. Complete the square:</p><p>u = 3x² + 9x + 7 = 3(x² + 3x) + 7 = 3(x + 3/2)² - 27/4 + 7</p><p>u = 3(x + 3/2)² + 1/4</p><p><strong>Step 4:</strong> Since 3(x + 3/2)² ≥ 0, we have u ≥ 1/4 (minimum at x = -3/2)</p><p><strong>Step 5:</strong> To maximize f(x) = 1 + 10/u, we need to maximize 10/u, which requires minimizing u.</p><p>When u is minimum (u = 1/4):</p><p>f(x)_max = 1 + 10/(1/4) = 1 + 40 = 41</p><p>∴ Maximum value of f(x) is <strong>41</strong></p>
Correct Answer: B