Question

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

• 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

Answer 1

Answer: (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 $\theta$
$\Rightarrow$ $\cos 2\theta +2\cos \theta \ge \frac{-3}{2}$ for all $\theta$
Also, maximum value of this expression is 3.