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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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Toolbox:
• 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
Step1
$sinx - cosx$ 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
edited Apr 4, 2013

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