Question

The minimum value of $\frac{x}{{\log }_{e}x}$ is

(a) e

(b) 1/e

(c) 1

(d) none of these

Answer 1

$$\begin{gathered} \left(a\right)e \\ \mathrm{Given}:f\left(x\right)=\frac{x}{{\log }_{e}x} \\ \Rightarrow f\text{'}\left(x\right)=\frac{{\log }_{e}x-1}{{\left({\log }_{e}x\right)}^2} \\ \mathrm{For}\mathrm{a}\mathrm{local}\mathrm{maxima}\mathrm{or}\mathrm{a}\mathrm{local}\mathrm{minima},\mathrm{we}\mathrm{must}\mathrm{have} \\ f\text{'}\left(x\right)=0 \\ \Rightarrow \frac{{\log }_{e}x-1}{{\left({\log }_{e}x\right)}^2}=0 \\ \Rightarrow {\log }_{e}x-1=0 \\ \Rightarrow {\log }_{e}x=1 \\ \Rightarrow x=e \\ \mathrm{Now}, \\ f\text{'}\text{'}\left(x\right)=\frac{-1}{x{\left({\log }_{e}x\right)}^2}+\frac{2}{x{\left({\log }_{e}x\right)}^3} \\ \Rightarrow f\text{'}\text{'}\left(e\right)=\frac{-1}{e}+\frac{2}{e}=\frac{1}{e}>0 \\ \mathrm{So},x=e\mathrm{is}\mathrm{a}\mathrm{local}\text{mini}\mathrm{ma}. \\ \therefore \mathrm{Minimum}\mathrm{value}\mathrm{of}f\left(x\right)=\frac{e}{{\log }_{e}e}=e \end{gathered}$$