Question:

Let $\mathbf{|M|}$ denote the determinant of a square matrix $\mathbf{M}$. Let $\mathbf{g: [0, \pi/2] \rightarrow \mathbb{R}}$ be the function defined by $\mathbf{g(\theta) = \sqrt{f(\theta)}^{-1} + \sqrt{f(\pi/2 - \theta)}^{-1}}$, where $\mathbf{f(\theta) = \frac{1}{2} |\begin{bmatrix} 1 & -\sin \theta \\ -\sin \theta & 1 \end{bmatrix}| + |\begin{bmatrix} \sin \theta & \cos (\theta + \pi/4) \\ \tan (\theta - \pi/4) & -\cos \pi/2 \end{bmatrix}|}$. Let $\mathbf{p(x)}$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $\mathbf{g(\theta)}$, and $\mathbf{p(2) = 2 - \sqrt{2}}$. Then, which of the following is/are TRUE?