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Question
If f (x) = ex sin x in [0, π], then c in Rolle's theorem is
Options
π/6
π/4
π/2
3π/4
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Solution
3π/4
The given function is
\[f'\left( x \right) = e^x \cos n x + \sin x e^x \]
\[ \Rightarrow f'\left( c \right) = e^c \cos c + \sin c e^c \]
\[\text{Now }, e^x cos x \text { is continuous and derivable in R } . \]
\[\text { Therefore, it is continuous on } \left[ 0, \pi \right] \text { and derivable on} \left( 0, \pi \right) . \]
\[ \therefore f'\left( c \right) = 0 \]
\[ \Rightarrow e^c \left( \cos c + \sin c \right) = 0\]
\[ \Rightarrow \left( \cos c + \sin c \right) = 0 \left( \because e^c \neq 0 \right)\]
\[ \Rightarrow \tan c = - 1\]
\[ \Rightarrow c = \frac{3\pi}{4}, \frac{7\pi}{4}, . . . \]
\[ \therefore c = \frac{3\pi}{4} \in \left( 0, \pi \right)\]
Thus, \[c = \frac{3\pi}{4} \in \left( 0, \pi \right)\]for which Rolle's theorem holds.
Hence, the required value of c is 3π/4.
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