Question

Let
$f\left(x\right)=\left|\begin{array}{ccc}1+{\sin }^{2}x& {\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& 1+{\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& {\cos }^{2}x& 1+4\sin 2x\end{array}\right|$
find the maximum value of $f\left(x\right)$

Answer 1

$f\left(x\right)=\left|\begin{array}{ccc}1+{\sin }^{2}x& {\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& 1+{\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& {\cos }^{2}x& 1+4\sin 2x\end{array}\right|$

$applying{C}_1\to C_1+C_{2}weget$

$f\left(x\right)=\left|\begin{array}{ccc}2& {\cos }^{2}x& 4\sin 2x\\ 2& 1+{\cos }^{2}x& 4\sin 2x\\ 1& {\cos }^{2}x& 1+4\sin 2x\end{array}\right|$

$applying{R}_2\to R_2-R_{1}and{R}_3\to R_3-R_1$

$f\left(x\right)=\left|\begin{array}{ccc}2& {\cos }^{2}x& 4\sin 2x\\ 2-2& 1+{\cos }^2-co{s}^{2}x& 4\sin 2x-4sin2x\\ 1-2& {\cos }^{2}x-co{s}^{2}x& 1+4\sin 2x-4sin2x\end{array}\right|$

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

$Sincethemaximumvalueof\sin 2xis1$

$\therefore themaximumvalueoff\left(x\right)is6$