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  4. If ' m ' and ' M ' represents the least and greatest values of the function f(θ)=(1/2 cos 2 θ-4 cos θ+6), then (M/m) is

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Q. 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

Trigonometric Functions

Solution:

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