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  4. Minimum positive integral value of ' k ' for which f(x)=k cos 2 x-4 cos 3 x has exactly one critical point in (0, π), is

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Q. Minimum positive integral value of ' $k$ ' for which $f(x)=k \cos 2 x-4 \cos ^{3} x$ has exactly one critical point in $(0, \pi)$, is

Application of Derivatives

Solution:

$f'(x) =-2 k \sin 2 x+12 \cos ^{2} x \sin x$
$=2 \sin 2 x(-k+3 \cos x)$
For $k \geq 3, x=\frac{\pi}{2}$ is the only critical point in $(0, \pi)$