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,$-1\le \sin x\le 1$
$\Rightarrow -\frac{5}{2}\le \sin x-\frac{3}{2}\le -\frac{1}{2}$
$\Rightarrow \frac{1}{4}\le {\left(\sin x-\frac{3}{2}\right)}^2\le \frac{25}{4}$
$\Rightarrow 1\le 4{\left(\sin x-\frac{3}{2}\right)}^2\le 25$
$\Rightarrow -1\le 4{\left(\sin x-\frac{3}{2}\right)}^2-2\le 23$