Functions

Let \(|M|\) denote the determinant of a square matrix \(M\). Let \(g:\left[0, \frac{\pi}{2}\right] \rightarrow \mathbb{R}\) be the function defined by

\(g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1}\)

where

$$ f(\theta)=\frac{1}{2}\left|\begin{array}{ccc} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \end{array}\right|+\left|\begin{array}{ccc} \sin \pi & \cos \left(\theta+\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\ \sin \left(\theta-\frac{\pi}{4}\right) & -\cos \frac{\pi}{2} & \log _{e}\left(\frac{4}{\pi}\right) \\ \cot \left(\theta+\frac{\pi}{4}\right) & \log _{e}\left(\frac{\pi}{4}\right) & \tan \pi \end{array}\right| . $$

Let \(p(x)\) be a quadratic polynomial whose roots are the maximum and minimum values of the function \(g(\theta)\), and \(p(2)=2-\sqrt{2}\). Then, which of the following is/are TRUE ?