Question

Write the point where f(x) = x log, x attains minimum value.

Answer 1

$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}f\left(x\right)=x{\log }_{e}x \\ \Rightarrow f\text{'}\left(x\right)={\log }_{e}x+1 \\ \mathrm{For}\mathrm{a}\mathrm{local}\mathrm{maxima}\mathrm{or}\mathrm{a}\mathrm{local}\mathrm{minima},\mathrm{we}\mathrm{must}\mathrm{have} \\ f\text{'}\left(x\right)=0 \\ \Rightarrow {\log }_{e}x+1=0 \\ \Rightarrow {\log }_{e}x=-1 \\ \Rightarrow x=\frac{1}{e} \\ \Rightarrow f\left(\frac{1}{e}\right)=\frac{1}{e}{\log }_e\left(\frac{1}{e}\right)=-\frac{1}{e} \\ \mathrm{Now}, \\ f\text{'}\text{'}\left(x\right)=\frac{1}{x} \\ \mathrm{At}x=\frac{1}{e}: \\ f\text{'}\text{'}\left(\frac{1}{e}\right)=\frac{1}{{\displaystyle \frac{1}{e}}}=e>0 \\ \mathrm{So},\left(\frac{1}{e},-\frac{1}{e}\right)\mathrm{is}\mathrm{a}\mathrm{point}\mathrm{of}\mathrm{local}\mathrm{minimum}. \end{gathered}$$.