# If the minimum and the maximum values of the function

Question:

If the minimum and the maximum values of the function

$f:\left[\frac{\pi}{4}, \frac{\pi}{2}\right] \rightarrow \mathrm{R}$, defined by

$f(\theta)=\left|\begin{array}{ccc}-\sin ^{2} \theta & -1-\sin ^{2} \theta & 1 \\ -\cos ^{2} \theta & -1-\cos ^{2} \theta & 1 \\ 12 & 10 & -2\end{array}\right|$ are $m$ and $M$

respectively, then the ordered pair $(m, M)$ is equal to :

1. (1) $(0,2 \sqrt{2})$

2. (2) $(-4,0)$

3. (3) $(-4,4)$

4. (4) $(0,4)$

Correct Option: , 2

Solution:

Applying $C_{2} \rightarrow C_{2}-C_{1}$

$f(\theta)=\left|\begin{array}{rrc}-\sin ^{2} \theta & -1 & 1 \\ -\cos ^{2} \theta & -1 & 1 \\ 12 & -2 & -2\end{array}\right|$

$=4\left(\cos ^{2} \theta-\sin ^{2} \theta\right)$

$=4 \cos 2 \theta, \theta \in\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$

Max. $f(\theta)=M=0$

Min. $f(\theta)=m=-4$

So, $(m, M)=(-4,0)$