Answer
$$\eqalign{
& {\text{Increasing on: }}\left( {\frac{{3\pi }}{2},\frac{{9\pi }}{2}} \right) \cr
& {\text{Decreasing on: }}\left( {0,\frac{{3\pi }}{2}} \right){\text{ and}}\left( {\frac{{9\pi }}{2},6\pi } \right) \cr
& f'\left( x \right) = - \cos \left( {\frac{x}{3}} \right) \cr} $$
Work Step by Step
$$\eqalign{
& f\left( x \right) = - 3\sin \left( {\frac{x}{3}} \right),{\text{ }}\left[ {0,6\pi } \right] \cr
& \left( {\text{a}} \right){\text{ Using a computer algebra system }}\left( {{\text{Geogrebra}}} \right){\text{ we obtain }} \cr
& {\text{the derivative of the function}}{\text{.}} \cr
& f'\left( x \right) = - 3\cos \left( {\frac{x}{3}} \right)\left( {\frac{1}{3}} \right) \cr
& f'\left( x \right) = - \cos \left( {\frac{x}{3}} \right) \cr
& \cr
& \left( {\text{b}} \right){\text{ Sketch the graph of }}f\left( x \right){\text{ and }}f'\left( x \right){\text{ }}\left( {{\text{See graph below}}} \right) \cr
& \left( {\text{c}} \right){\text{From the graph we obtain the critical numbers}} \cr
& x = \frac{{3\pi }}{2}{\text{ and }}x = \frac{{9\pi }}{2} \cr
& \cr
& \left( {\text{d}} \right){\text{ From the graph, we can see that the function is:}} \cr
& {\text{Increasing on: }}\left( {\frac{{3\pi }}{2},\frac{{9\pi }}{2}} \right) \cr
& {\text{Decreasing on: }}\left( {0,\frac{{3\pi }}{2}} \right){\text{ and}}\left( {\frac{{9\pi }}{2},6\pi } \right) \cr} $$
