The first derivative of a function $~g~$ is given by
$\qquad g\,^\prime(x)=\dfrac{\cos x}{2+\cos x}\,$.
Since $~g\,^\prime\left(\dfrac{\pi}{2}\right)=0\,$, it follows that $~\dfrac{\pi}{2}~$ is a critical number.
The value $~g\,^{\prime\prime}\left(\dfrac{\pi}{2}\right)$ is
[[☃ dropdown 1]], which makes $~g\left(\dfrac{\pi}{2}\right)~$ [[☃ dropdown 2]].