A) greater than\[-\frac{3}{2}\]
B) less than or equal to\[\frac{3}{2}\]
C) greater than or equal to \[\frac{-3}{2}\] and less than or equal to\[3\]
D) None of the above
Correct Answer: C
Solution :
We have, \[\cos 2\theta +2\cos \theta =2{{\cos }^{2}}\theta -1+2\cos \theta \] \[=2{{\left( \cos \theta +\frac{1}{2} \right)}^{2}}-\frac{3}{2}\] Now,\[2{{\left( \cos \theta +\frac{1}{2} \right)}^{2}}\ge 0\]for all\[\theta \] \[\therefore \] \[2{{\left( \cos \theta +\frac{1}{2} \right)}^{2}}-\frac{3}{2}\ge \frac{-3}{2}\]for all\[\theta \] \[\Rightarrow \] \[\cos 2\theta +2\cos \theta \ge \frac{-3}{2}\]for all\[\theta \] Also, maximum value of this expression is 3.
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