(a) Find the intervals in which the function f ( x )=(log x)/x is strictly increasing or strictly decreasing.

$$\mathrm{f}(\mathrm{x})=\frac{\log \mathrm{x}}{\mathrm{x}} \Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=\frac{1-\log \mathrm{x}}{\mathrm{x}^{2}} ; \mathrm{x}>0 $$ for strictly increasing/decreasing, put $\displaystyle \mathrm{f}^{\prime}(\mathrm{x})=0 \Rightarrow \mathrm{x}=e$ for strictly increasing, $\displaystyle \mathrm{x} \in(0, e)$ and for strictly decreasing $\displaystyle \mathrm{x} \in(e, \infty)$ $$\begin{aligned} & \mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{2}+\frac{2}{\mathrm{x}} ; \mathrm{x} \in[1,2] \\ & \Rightarrow \mathrm{f}^{I}(\mathrm{x})=\frac{1}{2}-\frac{2}{\mathrm{x}^{2}} \end{aligned} $$ for absolute maximum / minimum, put $\displaystyle \mathrm{f}^{\prime}(\mathrm{x})=0$ $\displaystyle \Rightarrow \mathrm{x}^{2}=4 \Rightarrow \mathrm{x}=2$ Now, $\displaystyle \mathrm{f}(1)=\frac{5}{2}$ and $\displaystyle \mathrm{f}(2)=2$ ∴ absolute maximum value $\displaystyle =\frac{5}{2}$ and absolute minimum value $\displaystyle =2$