Reduce the trigonometric arguments:

$\sin \left(\frac{7\pi }{2}-\theta \right)=-\cos \theta$, $\sin (11\pi +\theta )=-\sin \theta$, $\sin \left(\frac{3\pi }{2}-\theta \right)=-\cos \theta$, $\sin (9\pi -\theta )=\sin \theta$.

$f(\theta )=4({\cos }^4\theta +{\sin }^4\theta )-2({\cos }^6\theta +{\sin }^6\theta )$

Use ${\sin }^4+{\cos }^4=1-2{\sin }^2\theta {\cos }^2\theta$ and ${\sin }^6+{\cos }^6=1-3{\sin }^2\theta {\cos }^2\theta$:

$f=4(1-2s)-2(1-3s)=2-2s$ where $s={\sin }^2\theta {\cos }^2\theta =\frac{{\sin }^{2}2\theta }{4}$.

$f(\theta )=2-\frac{{\sin }^{2}2\theta }{2}$. So $\alpha =f_{\max }=2$ (at $\sin 2\theta =0$); $\beta =f_{\min }=\frac{3}{2}$ (at ${\sin }^{2}2\theta =1$).

$\alpha +2\beta =2+3=5$.