Question

25. 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

Answer: (D)

$4{\sin }^{2}x-12\sin x+7$ $=4({\sin }^{2}x-3\sin x)+7$ $=4\left[{\left(\sin x-\frac{3}{2}\right)}^2-\frac{9}{4}\right]+7$ $=4{\left(\sin x-\frac{3}{2}\right)}^2-2$ Since, $-1\le \sin x\le 1$ $\therefore$ $-\frac{5}{2}\le \sin x-\frac{3}{2}\le -\frac{1}{2}$ $\Rightarrow$ $\frac{1}{4}\le {\left(\sin x-\frac{3}{2}\right)}^2\le \frac{25}{4}$ $\Rightarrow$ $1\le 4{\left(\sin x-\frac{3}{2}\right)}^2\le 25$ $\Rightarrow$ $-1\le 4{\left(\sin x-\frac{3}{2}\right)}^2-2\le 23$