If $m$ and $M$ are the minimum and the maximum values of $4 +\large\frac{1}{2}$$\sin ^2 2x - 2 \cos ^4 x , x \in R$ then $M−m$ is equal to :

Question

$\begin{array}{1 1} (1) \large\frac{15}{4} \\ (2) \large\frac{9}{4} \\ (3) \large\frac{7}{4}\\ (4) \large\frac{1}{4} \end{array}$