Question

Check the validity of the Rolle's theorem for the following functions: $f\left(x\right)=e^{-x}\sin x,x\in [0,\pi ]$.

Answer 1

The function $e^{-x}$ and $\sin x$ are continuous and differentiable on their domains.

$\therefore f\left(x\right)=e^{-x}\sin x$ is continuous on $[0,\pi ]$

and differentiable on $(0,\pi )$

Now,

$f\left(0\right)=e^o\sin 0=1\times 0=0$

and

$f\left(\pi \right)=e^{-\pi }.\sin \pi =e^{-\pi }\times 0=0$

$\therefore f\left(0\right)=f\left(\pi \right)$

Thus, the function $f$ satisfies all the conditions of the Rolle's theorem.