Problem
Let |M| denote the determinant of a square matrix M. Let g: [0, π/2] → R be the function defined by g(θ) = √(f(θ) – 1) + √[f(π/2 – θ) – 1] where
f(θ) = \( \frac{1}{2} \begin{vmatrix} 1 & \sin{\theta} & 1 \\ -\sin{\theta} &1 &\sin{\theta}\\ -1& -\sin{\theta}&1 \\ \end{vmatrix} + \begin{vmatrix} \sin{\pi} & \cos{(\theta + \pi/4)} & \tan{(\theta – \pi/4)} \\ \sin{(\theta – \pi/4)} &-\cos{\pi/2} &\log_e{4/\pi}\\ \cot{(\theta + \pi/4)}& \log_e{\pi/4}& \tan{\pi} \\ \end{vmatrix} \)
Let p(x) be a quadratic polynomial whose roots are the maximum and minimum values of the function g(θ), and p(2) = 2 − √2. Then, which of the following is / are true?
A p(3 + √2) < 0
B p(1 + 3√2) > 0
C p((5√2 − 1)/4) > 0
D p((5 − √2)/4) < 0