Question

If ' $m$ ' and ' $M$ ' represents the least and greatest values of the function $f(\theta )=\frac{1}{2\cos 2\theta -4\cos \theta +6}$, then $\frac{M}{m}$ is

Answer 1

Answer: 4

$2\cos 2\theta -4\cos \theta +6=2\left(2{\cos }^2\theta -1\right)-4\cos \theta +6=4$ ${\cos }^2\theta -4\cos \theta +4=(2\cos \theta -1{)}^2+3$
$-3\le (2\cos \theta -1)\le 1$
$\Rightarrow 0\le (2\cos \theta -1{)}^2\le 9$
$\Rightarrow 3\le (2\cos \theta -1{)}^2+3\le 12$
$\Rightarrow \frac{1}{12}\le \frac{1}{(2\cos \theta -1{)}^2+3}\le \frac{1}{3}$
$\frac{M}{m}=\frac{1/3}{1/12}=4$