Find the intervals on which the function $\mathit{f}\left(\mathit{x}\right)={\left(\mathit{x}-1\right)}^3{\left(\mathit{x}-2\right)}^2$ is (a) strictly increasing (b) strictly decreasing.
OR
Find the dimensions of the rectangle of perimeter $36\mathrm{cm}$ which will sweep out a volume as large as possible, when revolved about one of its side. Also, find the maximum volume.