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Question
Verify Rolle's theorem for the following function on the indicated interval f(x) = cos 2x on [0, π] ?
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Solution
The given function is \[f\left( x \right) = \cos2x\] .
Since \[\cos2 \ x\] is everywhere continuous and differentiable.
Therefore, \[f\left( x \right)\] is continuous on \[\left[ 0, \pi \right]\] and differentiable on \[\left( 0, \pi \right)\] .
Also,\[f\left( \pi \right) = f\left( 0 \right) = 1\]
\[ \Rightarrow f'\left( x \right) = - 2\sin2x\]
\[ \Rightarrow - 2\sin2x = 0\]
\[ \Rightarrow \sin2x = 0\]
\[ \Rightarrow 2x = \pi\]
\[ \Rightarrow x = \frac{\pi}{2}\]
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