$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right)={\sin }^6\mathrm{x}+{\cos }^6\mathrm{x}\text{ for all real }\mathrm{x}\text{, then }\mathrm{f}\left(\mathrm{x}\right) \mathrm{lies} \mathrm{in} \mathrm{the} \mathrm{interval}$

$\begin{array}{l}f\left(x\right)={\cos }^{6}x+{\sin }^{6}x\\ ={\left({\cos }^{2}x+{\sin }^{2}x\right)}^3-3{\cos }^{2}x{\sin }^{2}x\left({\cos }^{2}x+{\sin }^{2}x\right)\\ =1-3{\cos }^{2}x\left(1-{\cos }^{2}x\right)\\ =3{\cos }^{4}x-3{\cos }^{2}x+1\\ =3\left({\cos }^{4}x-{\cos }^{2}x+\frac{1}{3}\right)\\ =3\left({\left({\cos }^{2}x-\frac{1}{2}\right)}^2+\frac{1}{12}\right)\\ \text{ Least value of }f\left(x\right)\text{ is }\frac{1}{4},\text{ when }{\cos }^{2}x-\frac{1}{2}=0\\ \text{ Greatest value of }f\left(x\right)\text{ is }1\text{, when }{\cos }^{2}x=0\text{ or }1\end{array}$