If $\alpha$ and $\beta$ are respectively the maximum and minimum values of the function f(x) given by

$f(x)=\left|\begin{array}{ccc}1+\sin ^2 x & \cos ^2 x & \sin 2 x \\ \sin ^2 x & 1+\cos ^2 x & \sin 2 x \\ \sin ^2 x & \cos ^2 x & 1+\sin 2 x\end{array}\right|$, then which one of the following options is not true?

a triangle can be constructed having its sides as $\alpha, \beta$ and $\alpha-\beta$

Applying $C_1 \rightarrow C+C_2$, we get

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

$=\left|\begin{array}{rrr} 2 & \cos ^2 x & \sin 2 x \\ 0 & 1 & 0 \\ -1 & 0 & 1 \end{array}\right|$        $\begin{array}{r}\text { Applying } R_2 \rightarrow R_2-R_1 \\ \text { and, } R_3 \rightarrow R_3-R_1\end{array}$

$=2+\sin 2 x$

We know that

$-1 \leq \sin 2 x \leq 1$ for all x

$\Rightarrow 1 \leq 2+\sin 2 x \leq 3$ for all x

$\Rightarrow 1 \leq f(x) \leq 3$ for all x

$\Rightarrow \alpha=3$ and $\beta=1$

Clearly, these values of $\alpha$ and $\beta$ satisfy (a), (b) and (c).