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The maximum value of $\frac{\operatorname{In} x}{x}$ is
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The correct answer is:
$\frac{1}{\mathrm{e}}$
$\mathrm{f}(\mathrm{x})=\frac{\ln \mathrm{x}}{\mathrm{x}}$
$f^{\prime}(x)=\frac{\left(\frac{1}{x}\right) x-\ell n x}{x^{2}}=\frac{1-\ell n x}{x^{2}}$
$\mathrm{f}^{\prime}(\mathrm{x})=0 \Rightarrow \frac{1-\ln \mathrm{x}}{\mathrm{x}^{2}}=0 \Rightarrow 1-\ell \mathrm{nx}=0$
$\Rightarrow \ell \mathrm{nx}=1 \quad \Rightarrow \mathrm{x}=\mathrm{e}$
$f^{\prime}(x)=\frac{\left(\frac{1}{x}\right) x-\ell n x}{x^{2}}=\frac{1-\ell n x}{x^{2}}$
$\mathrm{f}^{\prime}(\mathrm{x})=0 \Rightarrow \frac{1-\ln \mathrm{x}}{\mathrm{x}^{2}}=0 \Rightarrow 1-\ell \mathrm{nx}=0$
$\Rightarrow \ell \mathrm{nx}=1 \quad \Rightarrow \mathrm{x}=\mathrm{e}$
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