$\begin{array}{l}
y = - 2{\sin ^3}x + 3{\sin ^2}x + 6(2m - 1)\sin x + 2019\\
Dat\,\,t = \sin x,\,\,\,voi\,\,x \in \left( {\frac{\pi }{2};\frac{{3\pi }}{2}} \right)\,\,thi\,\,t\,\,giam\,\,tu\,\,1\,\,ve\,\, - 1\\
\Rightarrow y = - 2{t^3} + 3{t^2} + 6\left( {2m - 1} \right)t + 2019\,\,dong\,\,bien\,\,tren\,\,\left( { - 1;1} \right)\\
\Leftrightarrow y' = - 6{t^2} + 6t + 6\left( {2m - 1} \right) \ge 0 & \forall t \in \left( { - 1;1} \right)\\
\Leftrightarrow 2m - 1 \ge {t^2} - t\\
\Leftrightarrow m \ge \frac{{{t^2} - t + 1}}{2},\,\,\,\,\,\,\forall t \in \left( { - 1;1} \right)\\
\Leftrightarrow m \ge \mathop {\max \,\,}\limits_{\left( { - 1;1} \right)} \frac{{{t^2} - t + 1}}{2}\\
\Leftrightarrow m \ge \frac{3}{2}
\end{array}$
