Question

Let $f\left(x\right)=\left|\begin{array}{ccc}6-{\cos }^{2}x& {\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& 5+{\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& {\cos }^{2}x& 5+4\sin 2x\end{array}\right|,$ where $x\in R.$ If $M$ and $m$ denote the maximum and the minimum values of $f$ respectively, then the value of $M-m$ is

Answer 1

Given $f\left(x\right)=\left|\begin{array}{ccc}6-{\cos }^{2}x& {\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& 5+{\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& {\cos }^{2}x& 5+4\sin 2x\end{array}\right|$

Applying $R_1\to R_1-R_2$
$f\left(x\right)=\left|\begin{array}{ccc}5& -5& 0\\ {\sin }^{2}x& 5+{\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& {\cos }^{2}x& 5+4\sin 2x\end{array}\right|$

Applying $R_2\to R_2-R_3$
$f\left(x\right)=\left|\begin{array}{ccc}5& -5& 0\\ 0& 5& -5\\ {\sin }^{2}x& {\cos }^{2}x& 5+4\sin 2x\end{array}\right|$

$=25\left|\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ {\sin }^{2}x& {\cos }^{2}x& 5+4\sin 2x\end{array}\right|$

$f\left(x\right)=150+100\sin 2x$
$\therefore M=150+100=250$ and $m=150-100=50$