<p><strong>736.</strong> Let \(f(x) = (x-1)^{100}(x-2)^{2(99)}(x-3)^{3(98)}\cdots(x-100)^{100}\) where \(k = \dfrac{f'(101)}{f(101)}\). Find the value of \(\dfrac{k}{50} - 97\).</p>
Step-by-Step Solution
Key Concept: Use logarithmic differentiation: ln f(x) = 100·ln(x-1) + 2·99·ln(x-2) + 3·98·ln(x-3) + ... + 100·1·ln(x-100). Then f'(x)/f(x) = sum of [coefficient_k/(x-k)]. The coefficient of ln(x-k) is k(101-k), so evaluate at x=101.
<p><strong>Step 1:</strong> Identify the exponent pattern. The exponent of (x-k) is k·(101-k) for k=1,2,...,100.</p><p><strong>Step 2:</strong> Apply logarithmic differentiation:<br/>ln f(x) = Σ(k=1 to 100) k(101-k)·ln(x-k)</p><p><strong>Step 3:</strong> Differentiate:<br/>f'(x)/f(x) = Σ(k=1 to 100) k(101-k)/(x-k)</p><p><strong>Step 4:</strong> Evaluate at x = 101:<br/>k = f'(101)/f(101) = Σ(k=1 to 100) k(101-k)/100</p><p><strong>Step 5:</strong> Expand k(101-k) = 101k - k²<br/>k = (1/100)[101·∑k - ∑k²]<br/>k = (1/100)[101·(100·101/2) - (100·101·201/6)]<br/>k = (1/100)[511550 - 338350]<br/>k = 1732/10 = 173.2</p><p><strong>Step 6:</strong> Calculate k/50 - 97:<br/>k/50 - 97 = 173.2/50 - 97 = 3.464 - 97 = -93.536</p><p>∴ Answer: <strong>-93.536</strong> (or verify if answer expects 4900 from alternative calculation)</p>
Correct Answer: -93