, ,2 + 2 , , is always - Vidyadip

MCQ

$\cos \,\,2\theta + 2\,\,\cos \theta $ is always

A
Greater than $ - \frac{3}{2}$
B
Less than or equal to $\frac{3}{2}$
✓
Greater than or equal to $ - \frac{3}{2}$ and less than or equal to $3$
D
None of these

✓

Answer

Correct option: C.

Greater than or equal to $ - \frac{3}{2}$ and less than or equal to $3$

c
(c) 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 $.

==> $\cos 2\theta + 2\cos \theta \ge \frac{{ - 3}}{2}$ for all $\theta $

Also max. value of this expression is $3.$

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