The Range of  ${\sin }^{6}x+{\cos }^{6}x$

$f\left(x\right)={\sin }^{6}x+{\cos }^{6}x$$={\left({\sin }^{2}x\right)}^3+{\left({\cos }^{2}x\right)}^3$$=1-3{\sin }^{2}x{\cos }^{2}x$$=1-\frac{3}{4}.{\sin }^{2}2x$

$\left(\because {\sin }^{2}2x\in \left(0,1\right]\right)$

$$\begin{gathered} 0\le {\sin }^{2}2x\le 1 \\ \Rightarrow \frac{1}{4}\le 1-\frac{3}{4}{\sin }^{2}2x\le 1 \end{gathered}$$

$\underset{ }{minimum of f\left(x\right)}=\frac{1}{4}, maximum of \underset{ }{f\left(x\right)}=1$