If m and M are the minimum and the maximum values of 4 +1/2 sin²2x - 2 cos⁴x, x $\in$ R, then M - m is equal to

$\begin{array}{l}4+\frac{1}{2}\times 4(1-{\cos }^{2}x)-2{\cos }^{4}x\\ =4+2{\cos }^{2}x-2{\cos }^{4}x\\ =2(-2{\cos }^{4}x+{\cos }^{2}x+2)\\ =-4({\cos }^{4}x-\frac{{\cos }^{2}x}{2}-1)\\ =4\left[{({\cos }^{2}x-\frac{1}{4})}^2-\frac{17}{16}\right]\\ =\frac{17}{4}-4\left({({\cos }^{2}x-\frac{1}{4})}^2\right)\\ y_{Max}=M=\frac{17}{4}\\ y_{min}=m=\frac{17}{4}-(4\times \frac{9}{16})=\frac{8}{4}=2\\ Now M-m=\frac{17}{4}-2=\frac{9}{4}\end{array}$