Question

The maximum value of $4si{n}^{2}x-12sinx+7$ is.

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

Answer 1

The correct option is D None of these

$4si{n}^{2}x-12sinx+7$
$=4(si{n}^{2}x-3sinx)+7$
$=[{(sinx-\frac{3}{2})}^2-\frac{9}{4}]+7$
$=4{(sinx-\frac{3}{2})}^2-9+7$
$=4{(sinx-\frac{3}{2})}^2-2$
We know that , $-1\le sinx\le 1$
$\Rightarrow -\frac{5}{2}\le sinx-\frac{3}{2}\le -\frac{1}{2}$
$\Rightarrow \frac{1}{4}\le {(sinx-\frac{3}{2})}^2\le \frac{25}{4}$
$\Rightarrow 1\le 4{(sinx-\frac{3}{2})}^2\le 25$
$\Rightarrow -1\le 4{(sinx-\frac{3}{2})}^2-2\le 23$

Hence, maximum value is $23$.