Question

Let the maximum and minimum value of the expression $2cos^{2} \theta +cos⁡\theta +1$ is $M$ and $m$ respectively, then the value of $\left[\frac{M}{m}\right]$ is, (where [.] is the greatest integer function)

Answer 1

$2\left(cos\right)^{2} \theta +cos⁡\theta +1=2\left(\left(cos\right)^{2} ⁡ \theta + \frac{cos ⁡ \theta }{2} + \frac{1}{2}\right)$
$2\left\{\left(cos \theta + \frac{1}{4}\right)^{2} + \frac{7}{16}\right\}$
Given expression is maximum when $cos \theta =1$ and minimum when $cos \theta =-\frac{1}{4}$
$\Rightarrow M=2\left(\left(\frac{5}{4}\right)^{2} + \frac{7}{16}\right)=2\left(\frac{32}{16}\right)=4$
and $m=2\left(\frac{7}{16}\right)=\frac{7}{8}$
Hence, $\left[\frac{M}{m}\right]=\left[\frac{32}{7}\right]=4$