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Question
Verify Rolle's theorem for the following function on the indicated interval f(x) = 4sin x on [0, π] ?
Solution
The given function is \[f\left( x \right) = 4^{ sin \ x}\].
Since sine function and exponential function are everywhere continuous and differentiable, \[f\left( x \right)\] is continuous on \[\left[ 0, \pi \right]\] and differentiable on \[\left( 0, \pi \right)\] .
Also,
\[f\left( x \right) = 4^{sin \ x } \]
\[ \Rightarrow f'\left( x \right) = 4^{sin x} \left( \cos x \right)\log4\]
\[\therefore f'\left( x \right) = 0\]
\[ \Rightarrow 4^{sin x} \left( \cos x \right)\log4 = 0\]
\[ \Rightarrow 4^{ sin x } \cos x = 0\]
\[ \Rightarrow \cos x = 0\]
\[ \Rightarrow x = \frac{\pi}{2}\]
Thus, \[c = \frac{\pi}{2} \in \left( 0, \pi \right)\] such that \[f'\left( c \right) = 0\] .
Hence, Rolle's theorem is verified.
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