The maximum value of ${\sin }^{3}x. {\cos }^{3}x$ is ___ and is obtained at x =

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)={\sin }^3\mathrm{x}.{\cos }^3\mathrm{x} \\ ={\left(\sin \mathrm{x} \cos \mathrm{x}\right)}^3 \\ ={\left(\frac{2}{2} \sin \mathrm{x} \cos \mathrm{x}\right)}^3 \\ =\frac{1}{8} {\sin }^3 2\mathrm{x} \end{gathered}$$

$$\begin{gathered} \sin 2\mathrm{x} \mathrm{maximum} \mathrm{values} \mathrm{is} \\ \sin 2\mathrm{x}=1 \\ 2\mathrm{x}=\frac{\mathrm{\pi }}{2} \\ \mathrm{x}=\frac{\mathrm{\pi }}{4} \end{gathered}$$

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) \mathrm{maximum} \mathrm{value} \mathrm{is} \\ =\frac{1}{8} {\left(1\right)}^3=\frac{1}{8} \end{gathered}$$