If the maximum value of \( a \), for which the function \( f_a(x)=\tan^{-1}2x-3ax+7 \) is non-decreasing in \( \left(-\dfrac{\pi}{6},\dfrac{\pi}{6}\right) \), is \( \bar a \), then \( f_{\bar a}\!\left(\dfrac{\pi}{8}\right) \) is equal to:
\( 8-\dfrac{9\pi}{4(9+\pi^{2})} \)
\( 8-\dfrac{4\pi}{9(4+\pi^{2})} \)
\( 8\left(\dfrac{1+\pi^{2}}{9+\pi^{2}}\right) \)
\( 8-\dfrac{\pi}{4} \)
