If  $f\left(x\right)={\sin }^{6}x+{\cos }^{6}x$  for $n\in R$ , then $f\left(x\right)$  has

$\Rightarrow {\left({\sin }^{2}x\right)}^3+{\left({\cos }^{2}x\right)}^3={\left({\sin }^{2}x+{\cos }^{2}x\right)}^3-\left(3{\sin }^{2}x.{\cos }^{2}x\left({\sin }^{2}x+{\cos }^{2}x\right)\right)$

$\Rightarrow 1-3{\sin }^{2}x.{\cos }^{2}x$

$\Rightarrow 1-\frac{3\times 4{\sin }^{2}x.{\cos }^{2}x}{4}=1-\frac{3{\sin }^{2}2x}{4}$

$0\le {\sin }^{2}2x\le 1$  

$0\ge \frac{-3}{4}{\sin }^{2}2x\ge \frac{-3}{4}$

$0+1\ge 1-\frac{3}{4}{\sin }^{2}2x\ge 1-\frac{3}{4}$

$1\ge 1-\frac{3}{4}{\sin }^{2}2x\ge \frac{1}{4}$

$\therefore$  Maximum value is 1 and Minimum value is $\frac{1}{4}$