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Question
Verify Rolle's theorem for each of the following function on the indicated interval f (x) = cos 2 (x − π/4) on [0, π/2] ?
Solution
The given function is \[f\left( x \right) = \cos2\left( x - \frac{\pi}{4} \right) = \cos\left( 2x - \frac{\pi}{2} \right) = \sin2x\] .
Since \[\sin2 \ x \] is everywhere continuous and differentiable.
Therefore, \[\sin2x\] is continuous on \[\left[ 0, \frac{\pi}{2} \right]\] and differentiable on
\[ \Rightarrow f'\left( x \right) = 2\cos2x\]
\[ \Rightarrow 2\cos2x = 0\]
\[ \Rightarrow \cos2x = 0\]
\[ \Rightarrow x = \frac{\pi}{4}\]
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