Question

The maximum value of \(\frac{\log(x)}{x}\) is:

• A. \(\frac{2}{e}\)
• B. \(e\)
• C. 7
• D. \(\frac{1}{e}\)

Hint:

Optimization problems often require differentiation and identifying critical points.

Answer 1

The Correct Option is D

Let \( f(x) = \frac{\log(x)}{x} \). To find the maximum value, we first differentiate \( f(x) \) with respect to \(x\): \[ f'(x) = \frac{d}{dx}\left(\frac{\log(x)}{x}\right). \] Using the quotient rule: \[ f'(x) = \frac{x \cdot \frac{1}{x} - \log(x) \cdot 1}{x^2} = \frac{1 - \log(x)}{x^2}. \] Now, set \( f'(x) = 0 \) to find the critical points: \[ \frac{1 - \log(x)}{x^2} = 0 \quad \Rightarrow \quad 1 - \log(x) = 0 \quad \Rightarrow \quad \log(x) = 1 \quad \Rightarrow \quad x = e. \] To confirm that this is a maximum, check the second derivative or use the first derivative test. Substituting \(x = e\) into \( f(x) \): \[ f(e) = \frac{\log(e)}{e} = \frac{1}{e}. \] Thus, the maximum value is \(\frac{1}{e}\).