Applications of Derivatives
Absolute Maxima and Minima
Grade 12
Question:
<p>The function \(f(x) = \frac{4}{x - 1} - \frac{9}{x + 1}\) will</p>
<p>(a) have absolute maximum value -1/2</p>
<p>(b) have absolute minimum value -25/2</p>
<p>(c) have both absolute maximum and minimum values</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: Find critical points using f'(x) = 0 and evaluate the second derivative to determine which is a minimum.
<p>Let f(x) = 4/(x - 1) - 9/(x + 1)</p><p>Domain: x ≠ 1, -1</p><p>f'(x) = -4/(x - 1)² + 9/(x + 1)²</p><p>f'(x) = 0 gives: 9/(x + 1)² = 4/(x - 1)²</p><p>3/(x + 1) = ±2/(x - 1)</p><p>Case 1: 3(x - 1) = 2(x + 1) gives 3x - 3 = 2x + 2, so x = 5</p><p>Case 2: 3(x - 1) = -2(x + 1) gives 3x - 3 = -2x - 2, so 5x = 1, x = 1/5</p><p>At x = 5: f(5) = 4/4 - 9/6 = 1 - 3/2 = -1/2</p><p>At x = 1/5: f(1/5) = 4/(-4/5) - 9/(6/5) = -5 - 15/2 = -25/2</p><p>f''(x) analysis shows x = 1/5 is a local minimum with value -25/2</p><p>∴ Answer is (b) have absolute minimum value -25/2.</p>
Correct Answer: b