Find the Open Interval on Which the Curve Given by the Vector-Valued

Multiple Choice

Find the open interval on which the curve given by the vector-valued function $r ( \theta ) = ( 6 \theta + 6 \sin \theta ) \mathbf { i } + ( 8 - 6 \cos \theta ) \mathbf { j }$ is smooth.

A) $( ( 4 n - 8 ) \pi , ( 3 n + 2 ) \pi )$
B) $( ( 2 n - 8 ) \pi , ( 2 n + 8 ) \pi )$
C) $( ( 3 n - 1 ) \pi , ( 3 n + 1 ) \pi )$
D) $( ( 2 n - 1 ) \pi , ( 2 n + 1 ) \pi )$
E) $( ( 3 n - 8 ) \pi , ( 4 n + 2 ) \pi )$