The first derivative of a function $~g~$ is given by $\qquad g\,^\prime(x)=\dfrac{\tan^{-1}x}{x^2+1}\,$. It is clear that $~0~$ is a critical number because $~g\,^\prime(0)=0\,$. The value $~g\,^{\prime\prime}(0)~$ is [[☃ dropdown 1]], which makes $g(0)$ [[☃ dropdown 2]].