Mathematics - Application of Derivative Question with Solution | TestHub

Let $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{\log \mathrm{x}}$ $\Rightarrow {\mathrm{f}}^{\prime }(\mathrm{x})=\frac{\log \mathrm{x}-1}{(\log \mathrm{x}{)}^2}$ For maxima and minima, put $f^{\prime }(x)=0$ $\log x-1=0$ $\Rightarrow \mathrm{x}=\mathrm{e}$ Now, $f^{\prime \prime }(x)$ $=\frac{(\log x{)}^2\cdot \frac{1}{x}-(\log x-1)\cdot \frac{2\log x}{x}}{(\log x{)}^4}$ $\Rightarrow {\mathrm{f}}^{\prime \prime }(\mathrm{e})=\frac{\frac{1}{\mathrm{e}}-0}{1}=\frac{1}{\mathrm{e}}>0$ $\therefore \mathrm{f}(\mathrm{x})$ is minimum at $\mathrm{x}=\mathrm{e}$. Hence, minimum value of $f(x)$ at $x=e$ is $f(e)=\frac{e}{\log e}=e$