Answer
$$\eqalign{
& {\text{Increasing on }}\left( { - \pi , - \frac{{2\pi }}{3}} \right),\left( { - \frac{\pi }{3},0} \right),\left( {\frac{\pi }{3},\frac{{2\pi }}{3}} \right) \cr
& {\text{Decreasing on }}\left( { - \frac{{2\pi }}{3}, - \frac{\pi }{3}} \right),\,\left( {0,\frac{\pi }{3}} \right),\,\left( {\frac{{2\pi }}{3},\pi } \right) \cr} $$
Work Step by Step
$$\eqalign{
& f\left( x \right) = 3\cos 3x{\text{ on the interval }}\left[ { - \pi ,\pi } \right] \cr
& {\text{Derivative}} \cr
& f'\left( x \right) = 3\left( { - 3\sin 3x} \right) \cr
& f'\left( x \right) = - 9\sin 3x \cr
& {\text{Set the derivative to 0}} \cr
& - 9\sin 3x = 0 \cr
& {\text{Solving the equation on the interval }}\left[ { - \pi ,\pi } \right]{\text{ we obtain}} \cr
& x = - \frac{{2\pi }}{3},\,\,\, - \frac{\pi }{3},\,\,0,\,\,\,\frac{\pi }{3},\,\,\,\frac{{2\pi }}{3} \cr
& {\text{Now}}{\text{, we will evaluate the critical value and resume in a table}} \cr} $$
\[\begin{array}{*{20}{c}}
{{\rm{Interval}}}&{{\rm{Test value }}\left( x \right)}&{{\rm{Sign of }}f'\left( x \right)}&{{\rm{Behavior of }}f\left( x \right)}\\
{\left( { - \pi , - \frac{{2\pi }}{3}} \right)}&{ - \frac{{5\pi }}{6}}& + &{{\rm{Increasing}}}\\
{\left( { - \frac{{2\pi }}{3}, - \frac{\pi }{3}} \right)}&{ - \frac{\pi }{2}}& - &{{\rm{Decreasing}}}\\
{\left( { - \frac{\pi }{3},0} \right)}&{ - \frac{\pi }{6}}& + &{{\rm{Increasing}}}\\
{\left( {0,\frac{\pi }{3}} \right)}&{\frac{\pi }{6}}& - &{{\rm{Decreasing}}}\\
{\left( {\frac{\pi }{3},\frac{{2\pi }}{3}} \right)}&{\frac{\pi }{2}}& + &{{\rm{Increasing}}}\\
{\left( {\frac{{2\pi }}{3},\pi } \right)}&{\frac{{5\pi }}{6}}& - &{{\rm{Decreasing}}}
\end{array}\]
