Question

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

Options

• any interval

• the interval [0, π]

• the interval (0, π/2)

• none of these

Answer 1

 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 ​

\[\left[ 0, \pi \right]\].