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Question
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin 3x on [0, π] ?
Solution
The given function is \[f\left( x \right) = \sin3x\] .
Since \[\sin3x\] is everywhere continuous and differentiable, \[\sin3x\] is continuous on \[\left[ 0, \pi \right]\] and differentiable on \[\left( 0, \pi \right)\] .
Also,
\[ \Rightarrow f'\left( x \right) = 3\cos3x\]
\[ \Rightarrow 3\cos3x = 0\]
\[ \Rightarrow \cos3x = 0\]
\[ \Rightarrow 3x = \frac{\pi}{2}, \frac{3\pi}{2}, . . . . \]
\[ \Rightarrow x = \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}\]
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