Let the function \(f(\theta) = \begin{vmatrix} \sin\theta & \cos\theta &…

Let the function

$f(\theta) = \begin{vmatrix} \sin\theta & \cos\theta & \tan\theta \\ \sin(\frac{\pi}{6}) & \cos(\frac{\pi}{6}) & \tan(\frac{\pi}{6}) \\ \sin(\frac{\pi}{3}) & \cos(\frac{\pi}{3}) & \tan(\frac{\pi}{3}) \end{vmatrix}$

where $\theta \in \left[ \frac{\pi}{6},\frac{\pi}{3} \right]$ and  $f'(\theta)$ denote the derivative of $f$ with respect to $\theta$. Which of the following statements is/are TRUE?

I. There exists $\theta \in (\frac{\pi}{6},\frac{\pi}{3})$ such that $f'(\theta) = 0$

II.  There exists $\theta \in (\frac{\pi}{6},\frac{\pi}{3})$ such that $f'(\theta)\neq 0$