Logarithmic Differentiation in Optimization

00:00 6 marks

Given the function \(f(x) = x\ln(x) – x^2\), find its derivative, determine if there are any critical points for \(x > 0\), and explain why the function does not have a local maximum within its domain.

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Hint

Remember to use the product rule for the term \(x\ln(x)\). After finding the derivative, analyze its behavior to determine the existence of critical points and local maxima.

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The derivative is \(f'(x) = \ln(x) + 1 – 2x\). There are no interior critical points because \(f'(x) = \ln(x) + 1 – 2x = 0\) has no solution for \(x > 0\). The function \(f(x)\) is strictly decreasing for \(x > 0\) as \(f'(x) < 0\) for all \(x > 0\), hence no local maximum exists in the interior of the domain.

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