I have this function
$$
\begin{aligned}
F\big(\theta_a,\theta_b,\phi_a,\phi_b\big) = \ &
– \big[\cos \theta_a \cos \theta_b \big]
– \big[\sin\theta_a \sin\theta_b \sin\phi_a \sin\phi_b\big]
\\
& – \big[\cos\phi_a \cos\phi_b \sin\theta_a \sin\theta_b\big].
\end{aligned}
$$
I want to find the max values and the unknown variables $(\theta_a,\theta_b, \phi_a,\phi_b)$ analytically (by hand) and in MATLAB, the range for $\theta_a, \theta_b$ is $(0:\pi)$ and for $\phi_a, \phi_b$ is $(0:2 \pi)$.
Actually, the max value should be $2 \sqrt{2}$ that QM predicted to violate Bell's inequality.
As I tried in matlab I got $1$.
Any help will be valuable.Thanks
This is $-\left(\cos\theta_a\cos\theta_b+\sin\theta_a\sin\theta_b\cos(\phi_a-\phi_b)\right)$. Thus, the extreme values are taken for $\cos(\phi_a-\phi_b)=\pm1$, and they are $-\cos(\theta_a\pm\theta_b)$, so the minimum is $-1$ and the maximum is $1$.