If $\overrightarrow{\mathrm{a}}\text{ and }\overrightarrow{\mathrm{b}}$ are any two unit vectors, then the greatest positive integer in the range of $\frac{3|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|}{2}+2|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|$

 Let angle between $\overrightarrow{\mathrm{a}}\text{ and }\overrightarrow{\mathrm{b}}\text{ be }\mathrm{\theta }\text{. }$

$\text{ We have }\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=1$

$\text{ Now }|\overrightarrow{a}+\overrightarrow{b}|=2\cos \frac{\theta }{2}\text{ and }|\overrightarrow{a}-\overrightarrow{b}|=2\sin \frac{\theta }{2}$

Consider $\mathrm{F}\left(\mathrm{\theta }\right)=\frac{3}{2}\left(2\cos \frac{\mathrm{\theta }}{2}\right)+2\left(2\sin \frac{\mathrm{\theta }}{2}\right)$

$\therefore \mathrm{F}\left(\mathrm{\theta }\right)=3\cos \frac{\mathrm{\theta }}{2}+4\sin \frac{\mathrm{\theta }}{2},\mathrm{\theta }\in [0,\mathrm{\pi }]$