# If the minimum and the maximum values of the function

Question:

If the minimum and the maximum values of the function $\mathrm{f}:\left[\frac{\pi}{4}, \frac{\pi}{2}\right] \rightarrow \mathrm{R}$, defined by :

function $\mathrm{f}:\left[\frac{\pi}{4}, \frac{\pi}{2}\right] \rightarrow \mathrm{R}$, 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 $(\mathrm{m}, \mathrm{M})$ is equal to :

1. $(0,4)$

2. $(-4,4)$

3. $(0,2 \sqrt{2})$

4. $(-4,0)$

Correct Option: , 4

Solution:

$\mathrm{C}_{3} \rightarrow \mathrm{C}_{3}-\left(\mathrm{C}_{1}-\mathrm{C}_{2}\right)$

$f(\theta)=\left|\begin{array}{ccc}-\sin ^{2} \theta & -1-\sin ^{2} \theta & 0 \\ -\cos ^{2} \theta & -1-\cos ^{2} \theta & 0 \\ 12 & 10 & -4\end{array}\right|$

$=-4\left[\left(1+\cos ^{2} \theta\right) \sin ^{2} \theta-\cos ^{2} \theta\left(1+\sin ^{2} \theta\right)\right]$

$=-4\left[\sin ^{2} \theta+\sin ^{2} \theta \cos ^{2} \theta-\cos ^{2} \theta-\cos ^{2} \theta \sin ^{2} \theta\right]$

$f(\theta)=4 \cos 2 \theta$

$\theta \in\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$

$2 \theta \in\left[\frac{\pi}{2}, \pi\right]$

$f(\theta) \in[-4,0]$

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