The minimum value of $\mathrm{f}\left(\mathrm{x}\right)=2^{\left({\log }_{8}3\right){\cos }^2\mathrm{x}}+3^{\left({\log }_{8}2\right){\sin }^2\mathrm{x}}$ is

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=3^{\left({\log }_{8}2\right){\cos }^2\mathrm{x}}+3^{\left({\log }_{8}2\right){\sin }^2\mathrm{x}} \\ \mathrm{f}\left(\mathrm{x}\right)=3^{\frac{1}{3}\left({\log }_{2}2\right){\cos }^2\mathrm{x}}+3^{\frac{1}{3}\left({\log }_{2}2\right)\left(1-{\cos }^2\mathrm{x}\right)} \\ \mathrm{f}\left(\mathrm{x}\right)=3^{\frac{{\cos }^2\mathrm{x}}{3}}+\frac{3^{\frac{1}{3}}}{3^{\frac{{\cos }^2\mathrm{x}}{3}}}\Rightarrow \text{ Min. }=2\sqrt{3^{\frac{1}{3}}}=2\times 3^{\frac{1}{6}}=2^{1+{\log }_8\sqrt{3}} \end{gathered}$$