Question

The maximum value of $4 \, \sin^2 \, x - 12 \sin \, x + 7$ is

• A. 25
• B. 4
• C. does not exist
• D. None of these

Answer 1

Answer: (D)

$4 \sin^{2} x = 12\sin x+7 $
$=4\left(\sin^{2}x -3\sin x\right) + 7 $
$= 4\left[\left(\sin x - \frac{3}{2}\right)^{2} - \frac{9}{4}\right]+7$
$ =4\left(\sin x - \frac{3}{2}\right)^{2} -9 + 7$
$ = 4 \left(\sin x - \frac{3}{2} \right)^{2} - 2$
We know that,$ - 1 \le\sin x \le1$
$ \Rightarrow - \frac{5}{2} \le\sin x - \frac{3}{2}\le- \frac{1}{2} $
$\Rightarrow \frac{1}{4}\le\left(\sin x - \frac{3}{2}\right)^{2} \le \frac{25}{4}$
$ \Rightarrow 1 \le4 \left(\sin x -\frac{3}{2}\right)^{2} \le25$
$ \Rightarrow - 1 \le4 \left(\sin x - \frac{3}{2}\right)^{2} - 2 \le23 $