A) \[\left\{ \frac{\pi }{2} \right\}\cup \left[ \frac{\pi }{6},\,\,\frac{5\pi }{6} \right]\]
B) \[\left\{ \frac{\pi }{2} \right\}\cup \left[ 0,\,\,\frac{\pi }{6} \right]\]
C) \[[0,\,\,\pi ]\]
D) \[\left\{ \frac{\pi }{2} \right\}\cup \left[ 0,\,\,\frac{\pi }{6} \right]\cup \left[ \frac{5\pi }{6},\,\,\pi \right]\]
Correct Answer: D
Solution :
| \[|\cos 3x|+|\cos x|\,\,=\,\,|\cos 3x+\cos x|\] |
| if \[\cos 3x\cdot \cos x\ge 0\] |
| if \[(4{{\cos }^{3}}x-\cos x)\cos \,\,x\ge 0\] |
| if \[{{\cos }^{2}}x\,\,(4{{\cos }^{2}}x-3)\ge 0\] |
| if \[\cos x=0\] or \[{{\cos }^{2}}x\ge \frac{3}{4}\] |
| if \[x=\frac{\pi }{2}\] or \[|\cos x|\,\,\ge \frac{\sqrt{3}}{2}\] |
 |
| \[x=\frac{\pi }{2}\] or \[x\in \left[ 0,\,\,\frac{\pi }{6} \right]\cup \left[ \frac{5\pi }{6},\,\,\pi \right]\] |
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