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Verify Rolle’s theorem for the function $f(x)=\sin 2x \;in\;\bigg[0,\Large\frac{\pi}{2}\bigg]$

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Toolbox:
  • Check $ f $ is continuous and differentiable in $ \bigg[ 0, \large\frac{\pi}{2} \bigg]$ and $ f(0)=f\bigg( \large\frac{\pi}{2} \bigg)$
  • Then find C
$ f$ is continuous and differentiable.
$ f(0)=f\bigg( \large\frac{\pi}{2}\bigg) = a$
$ \Rightarrow \exists $ a point $ C \in \bigg( 0,\large \frac{\pi}{2} \bigg)$
 
$ f ' (C)=0$
$ f ' (x)=2cos2x = 0$
$ \Rightarrow x = \large\frac{\pi}{4} \in \bigg( 0,\large \frac{\pi}{2} \bigg)$
$ C = \large\frac{\pi}{4}$

 

answered Mar 12, 2013 by thanvigandhi_1
edited Mar 26, 2013 by thanvigandhi_1
 

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