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Question
Rolle's theorem is applicable in case of ϕ (x) = asin x, a > a in
Options
any interval
the interval [0, π]
the interval (0, π/2)
none of these
Solution
the interval [0, π]
The given function is \[\phi\left( x \right) = a^{sin x}\], where a > 0.
Differentiating the given function with respect to x, we get
\[f'\left( x \right) = \log a\left( \cos x a^{sin x } \right)\]
\[\Rightarrow f'\left( c \right) = \log a\left( \cos c \ a^{sin c} \right)\]
\[Let f'\left( c \right) = 0 \]
\[ \Rightarrow \log a\left( \cos c a^{sin \ c } \right) = 0\]
\[ \Rightarrow \cos c a^{sin \ c} = 0\]
\[ \Rightarrow \cos c = 0\]
\[ \Rightarrow c = \frac{\pi}{2}\]
∴ \[c \in \left( 0, \pi \right)\]
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
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