Let the function
\[{\text{f}}\left( \theta \right) = \left| {\begin{array}{*{20}{c}} {\sin \theta }&{\cos \theta }&{\tan \theta } \\ {\sin \left( {\frac{\pi }{6}} \right)}&{\cos \left( {\frac{\pi }{6}} \right)}&{\tan \left( {\frac{\pi }{6}} \right)} \\ {\sin \left( {\frac{\pi }{3}} \right)}&{\cos \left( {\frac{\pi }{3}} \right)}&{\tan \left( {\frac{\pi }{3}} \right)} \end{array}} \right|\]
where \[\theta \in \left[ {\frac{\pi }{6},\,\frac{\pi }{3}} \right]\]   and \[{\text{f'}}\left( \theta \right)\]  denote the derivative of f with respect to \[\theta \]. Which of the following statements is/are TRUE?
I. There exists \[\theta \in \left( {\frac{\pi }{6},\,\frac{\pi }{3}} \right)\]   such that \[{\text{f'}}\left( \theta \right) = 0.\]
II. There exists \[\theta \in \left( {\frac{\pi }{6},\,\frac{\pi }{3}} \right)\]   such that \[{\text{f'}}\left( \theta \right) \ne 0\]