Consider the function \(f(x) = \ln(2x) – x\) defined for \(x > 0\).
(a) Express \(f(x)\) in a form that separates the constant from the variable-dependent logarithm using logarithmic properties.
(b) Differentiate \(f(x)\) with respect to \(x\).
(c) Determine the value of \(x\) that maximizes \(f(x)\) and verify that this critical point corresponds to a maximum.
Question
Rewrite \(f(x) = \ln(2x) – x\) using the identity \(\ln(ab) = \ln(a) + \ln(b)\) to separate the constant term from the logarithmic term involving \(x\).
Hint
Hint
Hint
Separate \(\ln(2x)\) into \(\ln(2)\) and \(\ln(x)\).
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- Formula Booklet
Complete Answer
\(f(x) = \ln(2) + \ln(x) – x\)
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Question
Differentiate the reformulated function \(f(x) = \ln(2) + \ln(x) – x\) with respect to \(x\).
Hint
Hint
Hint
Recall that the derivative of a constant is 0 and that \(\frac{d}{dx}[\ln(x)] = \frac{1}{x}\).
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Answer
Answer
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- Formula Booklet
Complete Answer
\(f'(x) = \frac{1}{x} – 1\)
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Question
Determine the value of \(x\) that maximizes \(f(x)\) by setting \(f'(x) = 0\), and verify the result using the second derivative test.
Hint
Hint
Hint
Set \(\frac{1}{x} – 1 = 0\) to solve for \(x\) and then compute \(f”(x)\).
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
\(x = 1\)
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