Rolle’s theorem is applicable in case
Question:

Rolle’s theorem is applicable in case of $\phi(x)=a^{\sin x}, a>a$ in

(a) any interval
(b) the interval [0, π]
(c) the interval (0, π/2)
(d) none of these

Solution:

(b) the interval $[0, \pi]$

The given function is $\phi(x)=a^{\sin x}$, where $a>0$.

Differentiating the given function with respect to x, we get

$f^{\prime}(x)=\log a\left(\cos x a^{\sin x}\right)$

$\Rightarrow f^{\prime}(c)=\log a\left(\cos c a^{\sin c}\right)$

Let $f^{\prime}(c)=0$

$\Rightarrow \log a\left(\cos c a^{\sin c}\right)=0$

$\Rightarrow \cos c a^{\sin c}=0$

$\Rightarrow \cos c=0$

$\Rightarrow c=\frac{\pi}{2}$

$\therefore c \in(0, \pi)$

Also, the given function is derivable and hence continuous on the interval $[0, \pi]$.

Hence, the Rolle’s theorem is applicable on the given function in the interval $[0, \pi]$.