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Q.
The difference between the greatest and the least value of $f(x)=\cos ^2 \frac{x}{2} \sin x, x \in[0, \pi]$ is
Application of Derivatives
Solution:
$ f ( x )=\frac{\sin x (1+\cos x )}{2} ; f ^{\prime}( x )=\frac{1}{2}(\cos x +\cos 2 x ) \Rightarrow \cos x +2 \cos ^2 x -1=0 $
$\Rightarrow 2 \cos ^2 x +\cos x -1=0 \Rightarrow 2 \cos ^2 x +2 \cos x -\cos x -1=0 $
$\Rightarrow 2 \cos x (\cos x +1)-(1+\cos x )=0 \Rightarrow \cos x =1 / 2 \text { or } \cos x =-1 $
$\Rightarrow x =\pi \text { or } \frac{\pi}{3}$