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Verify the Rolle’s Theorem for the function $ f(x)=\sin x- \cos x, $in the interval $ \bigg[\Large\frac{\pi}{4},\frac{5\pi}{4}\bigg]$

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  • Check whether the function is continuous or not in the given closed interval [ a, b ].
  • Check whether it is differentiable or not in the given open interval ( a, b ).
  • Check whether f(a) = f(b)
  • Then find C in (a, b) / f'(c)=0
$ sinx - cosx$ is continuous and differentiable everywhere.
Step 2
$ f\bigg( \large\frac{\pi}{4} \bigg) = sin\large\frac{\pi}{4}-cos\large\frac{\pi}{4}=0$
$ f \bigg(\large \frac{5\pi}{4} \bigg) = sin\large\frac{5\pi}{4}-cos\large\frac{5\pi}{4}=0$
Step 3
Then as per $ R.T \: \: \exists$ atleast one point say $ 'c' \in \bigg( \large\frac{\pi}{4} \large\frac{5\pi}{4} \bigg) $ such that $ f'(c)=0$
$ f'(x)=cos x + sin x = 0$
Step 4
$ \Rightarrow x =\large \frac{3\pi}{4} \in \bigg( \large\frac{\pi}{4} \large\frac{5\pi}{4} \bigg) c= \large\frac{3\pi}{4}$


answered Mar 8, 2013 by thanvigandhi_1
edited Apr 4, 2013 by thanvigandhi_1

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