The function $\mathrm{f} : \mathrm{R}\to \mathrm{R}$ is defined $\mathrm{f}\left(\mathrm{x}\right)={\cos }^2 \mathrm{x}+{\sin }^4 \mathrm{x}$ for $\mathrm{x}\in \mathrm{R}$. Then the range of f(x) is

$\begin{array}{l}\mathrm{y}=\mathrm{f}\left(\mathrm{x}\right)={\cos }^2\mathrm{x}+{\sin }^4\mathrm{x}\\ ={\cos }^2\mathrm{x}+{\sin }^2\mathrm{x}\left(1-{\cos }^2\mathrm{x}\right)\\ ={\cos }^2\mathrm{x}+{\sin }^2\mathrm{x}-{\sin }^2{\mathrm{xcos}}^2\mathrm{x}\\ =1-{\sin }^2{\mathrm{xcos}}^2\mathrm{x}\\ =1-\frac{1}{4}{\sin }^{2}2\mathrm{x}\end{array}$

$\begin{array}{l}\therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\frac{3}{4}\le \mathrm{f}\left(\mathrm{x}\right)\le 1\left(\because 0\le {\sin }^{2}2\mathrm{x}\le 1\right)\\ \therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{f}\left(\mathrm{x}\right)\in [3/4,1]\end{array}$