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 the above

Answer 1

The correct option is D None of the above

$4{\sin }^{2}x-12\sin x+7$
$=4({\sin }^{2}x-3\sin x)+7$
$=4[{(\sin x-\frac{3}{2})}^2-\frac{9}{4}]+7$

$=4{(\sin x-\frac{3}{2})}^2-9+7$
$=4{(\sin x-\frac{3}{2})}^2-2$
Since, $-1\le \sin x\le 1$
$\therefore -\frac{5}{2}\le \sin x-\frac{3}{2}\le -\frac{1}{2}$
$\Rightarrow \frac{1}{4}\le {(\sin x-\frac{3}{2})}^2\le \frac{25}{4}$ ..... [Squaring both sides]
$\Rightarrow 1\le 4{(\sin x-\frac{3}{2})}^2\le 25$
$\Rightarrow -1\le 4{(\sin x-\frac{3}{2})}^2-2\le 23$

Hence, the maximum value of $4{\sin }^{2}x-12\sin x+7$ is $23$.