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📅 March 12, 2026 👤 scraface

Mar 12, 2026

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📌 A health data analyst models patient recovery rates with the function \( R(x) = \ln(x^2 + 4x + 5) \). Find the value of \( x \) that maximizes \( R(x) \).

📌 To maximize \( R(x) = \ln(x^2 + 4x + 5) \), we first find the derivative \( R'(x) \) and set it to zero.

📌 Differentiate \( R(x) \):

📌 \[ R'(x) = \frac{d}{dx}[\ln(x^2 + 4x + 5)] = \frac{2x + 4}{x^2 + 4x + 5} \]

📌 Set the derivative to zero:

📌 \[ \frac{2x + 4}{x^2 + 4x + 5} = 0 \]

📌 The numerator must be zero, so:

📌 \[ 2x + 4 = 0 \]

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