Question

Rolle's theorem is applicable in case of ϕ (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

Answer 1

(b) the interval [0, π]1−5√2

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

Differentiating the given function with respect to x, we get

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

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

$$\begin{gathered} \mathrm{Let}f\text{'}\left(c\right)=0 \\ \Rightarrow \log a\left(\cos c{a}^{\sin c}\right)=0 \\ \Rightarrow \cos c{a}^{\sin c}=0 \\ \Rightarrow \cos c=0 \\ \Rightarrow c=\frac{\mathrm{\pi }}{2} \end{gathered}$$
∴ $c\in \left(0,\mathrm{\pi }\right)$

Also, the given function is derivable and hence continuous on the interval $\left[0,\mathrm{\pi }\right]$.

Hence, the Rolle's theorem is applicable on the given function in the interval ​$\left[0,\mathrm{\pi }\right]$.