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  3. Find the local maxima and local minima, if any , of the following function.Also, find the local maximun and local minimum values as the case maybe:
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Find the local maxima and local minima, if any , of the following function.Also, find the local maximun and local minimum values as the case maybe:

f(x)=\sin x+\frac{1}{2}\cos 2x, 0\leq x\leq \frac{\pi}{2}

 

 

 

 
 
 
 
 

Answers (1)
R Ravindra Pindel

f(x)=\sin x+\frac{1}{2}\cos x, 0\leq x\leq \frac{\pi}{2}

f'(x)=\cos x - \sin 2x =0

\cos x =\sin 2x

\cos x =2\sin x\cdot \cos x

1 =2\sin x

\sin x=1/2 .........(1)

Differentiating again,

f''(x)=- \sin x-2\cos 2x

=-1/2-1=-3/2< 0

x=\frac{\pi}{6}\Rightarrow local\: maxima

f\left ( \frac{\pi}{6} \right )= \sin \frac{x}{6}+\frac{1}{2}\cos 2 \times \frac{\pi}{6}

=\frac{1}{2}+\frac{1}{2}.\frac{1}{2}

=\frac{3}{4}  ...........Local Maxima.

 

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