Mathematics - Functions Question with Solution | TestHub

$|\le |\sin x|+|\cos x|\le \sqrt{2}$

$[|\sin x|+|\cos x|]=1\forall x\in \mathbb{R}$

Now ${\sin }^{2}x+2\sin x+\frac{11}{4}=(\sin x+1{)}^2+\frac{7}{4}$

For f to be well defined $(\sin x+1{)}^2+\frac{7}{4}\ge 2$ $(\sin x+1{)}^2\ge \frac{1}{4}$ $\Rightarrow \sin x+1\ge \frac{1}{2},\sin x+1\le -\frac{1}{2}$ $\sin x\ge -\frac{1}{2},\sin x\le -\frac{3}{2}$ (This is impossible $\Rightarrow x\in \left[0,\frac{7\pi }{6}\right]U\left[\frac{11\pi }{6},2\pi \right]$ Hence (A) is correct