Question

Write the maximum value of f(x) = $\frac{\log x}{x}$, if it exists.

Answer 1

$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}f\left(x\right)=\frac{\log x}{x} \\ \Rightarrow f\text{'}\left(x\right)=\frac{1-\log x}{x^2} \\ \mathrm{For}\mathrm{a}\mathrm{local}\mathrm{maxima}\mathrm{or}\mathrm{a}\mathrm{local}\mathrm{minima},\mathrm{we}\mathrm{must}\mathrm{have} \\ f\text{'}\left(x\right)=0 \\ \Rightarrow \frac{1-\log x}{x^2}=0 \\ \Rightarrow 1-\log x=0 \\ \Rightarrow \log x=1 \\ \Rightarrow \log x=\log e \\ \Rightarrow x=e \\ \mathrm{Now}, \\ f\text{'}\text{'}\left(x\right)=\frac{-x-2x\left(1-\log x\right)}{x^4}=\frac{-3x-2x\log x}{x^4} \\ \mathrm{At}x=e: \\ f\text{'}\text{'}\left(e\right)=\frac{-3e-2e\log e}{e^4}=\frac{-5}{e^3}<0 \\ \mathrm{So},x=e\mathrm{is}\mathrm{a}\mathrm{point}\mathrm{of}\mathrm{local}\mathrm{maximum}. \\ \mathrm{Thus},\mathrm{the}\mathrm{local}\mathrm{maximum}\mathrm{value}\mathrm{is}\mathrm{given}\mathrm{by} \\ f\left(e\right)=\frac{\log e}{e}=\frac{1}{e} \end{gathered}$$.