Rolle's theorem is applicable in case of ϕ (x) = asin x, a a in

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Rolle's theorem is applicable in case of ϕ (x) = asin x, a > a in

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the interval [0, $\pi$] The given function is \phi\left( x \right) = a^{sin x}, where a > 0. Differentiating the given function with respect to x, we get f'\left( x \right) = \log a\left( \cos x a^{sin x } \right) \Rightarrow f'\left( c \right) = \log a\left( \cos c \ a^{sin c} \right) Let f'\left( c \right) = 0 \Rightarrow \log a\left( \cos c a^{sin \ c } \right) = 0 \Rightarrow \cos c a^{sin \ c} = 0 \Rightarrow \cos c = 0 \Rightarrow c = $\pi$/2 $\therefore$ c \in \left( 0, $\pi$ \right) Also, the given function is derivable and hence continuous on the interval \left[ 0, $\pi$ \right]. Hence, the Rolle's theorem is applicable on the given function in the interval ​ \left[ 0, $\pi$ \right].

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