Question

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

Answer 1

Answer: 4

$2{\left(cos\right)}^2\theta +cos\theta +1=2\left({\left(cos\right)}^2\theta +\frac{cos\theta }{2}+\frac{1}{2}\right)$
$2\left\{{\left(cos\theta +\frac{1}{4}\right)}^2+\frac{7}{16}\right\}$
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, $\left[\frac{M}{m}\right]=\left[\frac{32}{7}\right]=4$