The maximum value of $4{\sin }^{2}x+3{\cos }^{2}x+\sin \frac{x}{2}+\cos \frac{x}{2} is$

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=4{\sin }^2\mathrm{x}+3{\cos }^2\mathrm{x}+\sin \frac{x}{2}+\cos \frac{x}{2} \\ ={\sin }^2\mathrm{x}+3+\sin \frac{x}{2}+\cos \frac{x}{2} \end{gathered}$$

maximum value of ${\sin }^2\mathrm{x}=1$

and $\sin \frac{x}{2}+\cos \frac{x}{2}$ maximum value is $0+\sqrt{1+1}=\sqrt{2}$

maximum of $\mathrm{f}\left(\mathrm{x}\right)$ is $1+3+\sqrt{2}$

$=4+\sqrt{2}$