Let M and m respectively be the maximum and the minimum values of $\mathrm{f}\left(\mathrm{x}\right)=\left|\begin{array}{ccc}1+{\sin }^2\mathrm{x}& {\cos }^2\mathrm{x}& 4\sin 4\mathrm{x}\\ {\sin }^2\mathrm{x}& 1+{\cos }^2\mathrm{x}& 4\sin 4\mathrm{x}\\ {\sin }^2\mathrm{x}& {\cos }^2\mathrm{x}& 1+4\sin 4\mathrm{x}\end{array}\right|,\mathrm{x}\in \mathrm{R}$ Then M⁴ – m⁴ is equal to :

$\begin{array}{l}\left|\begin{array}{ccc}1+{\sin }^2\mathrm{x}& {\cos }^2\mathrm{x}& 4\sin 4\mathrm{x}\\ {\sin }^2\mathrm{x}& 1+{\cos }^2\mathrm{x}& 4\sin 4\mathrm{x}\\ {\sin }^2\mathrm{x}& {\cos }^2\mathrm{x}& 1+4\sin 4\mathrm{x}\end{array}\right|,\mathrm{x}\in \mathrm{R}\\ {\mathrm{R}}_2\to {\mathrm{R}}_2-{\mathrm{R}}_1\mathrm{\&}{\mathrm{R}}_3\to {\mathrm{R}}_3\to {\mathrm{R}}_1\\ \left(\mathrm{x}\right)\left|\begin{array}{lcc}1+{\sin }^2\mathrm{x}& {\cos }^2\mathrm{x}& 4\sin 4\mathrm{x}\\ -1& 1& 0\\ -1& 0& 1\end{array}\right|\end{array}$
Expand about R₁, use get
f(x) = 2 + 4sin4x
$\therefore$ M = max value of f(x) = 6
M = min value of f(x) = –2
$\therefore$ M⁴ – M⁴ = 1280