Question

Let the maximum and minimum value of the expression $2{\cos }^2\theta +\cos \theta +1$ 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

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