Show that $\displaystyle \mathrm{f}(x)=\frac{4 \sin x}{2+\cos x}-x$ is an increasing function of $\displaystyle x$ in $\displaystyle \left[0, \frac{\pi}{2}\right]$.

$$\begin{aligned} & \mathrm{f}(x)=\frac{4 \sin x}{2+\cos x}-x \Rightarrow \mathrm{f}^{\prime}(x)=\frac{\cos x(4-\cos x)}{(2+\cos x)^{2}} \\ & \text { when } x \in\left[0, \frac{\pi}{2}\right], \cos x \geq 0 \Rightarrow \frac{\cos x(4-\cos x)}{(2+\cos x)^{2}} \geq 0 \\ & \text { Thus, } \mathrm{f} \text { is increasing on }\left[0, \frac{\pi}{2}\right] . \end{aligned} $$