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प्रश्न
Verify Rolle's theorem for the following function on the indicated interval f(x) = ex sin x on [0, π] ?
उत्तर
The given function is \[f\left( x \right) = e^x \sin x\] .
Since\[\text { sin } x \text { and } e^{x} \] are everywhere continuous and differentiable.
Therefore, being a product of these two, \[f\left( x \right)\]is continuous on \[\left[ 0, \pi \right]\] and differentiable on \[\left( 0, \pi \right)\] .
Also,
Thus,
\[f\left( x \right)\] satisfies all the conditions of Rolle's theorem.
Now, we have to show that there exists\[c \in \left( 0, \pi \right)\] such that \[f'\left( c \right) = 0\] .
We have
\[f\left( x \right) = e^x \sin x\]
\[ \Rightarrow f'\left( x \right) = e^x \left( \sin x + \cos x \right)\]
\[\therefore f'\left( x \right) = 0\]
\[ \Rightarrow e^x \left( \sin x + \cos x \right) = 0\]
\[ \Rightarrow \sin x + \cos x = 0\]
\[ \Rightarrow \tan x = - 1\]
\[ \Rightarrow x = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\]
Since
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