Question

Verify Rolle's theorem for each of the following functions on the indicated intervals:
$f\left(x\right)=e^x\sin x$ on $[0,\pi ]$.

Answer 1

We need to verify Rolle's theorem for $f\left(x\right)=e^{-x}sinx$ on $[0,\pi ]$

$e^{-x}$ and $sinx$ are continuous for all $x,$therefore the product $e^{-x}sinx$ is continuous in $0\le x\le \pi$

$f^{\prime }\left(x\right)=-e^{-x}sinx+e^{-x}cosx=e^{-x}(cosx-sinx)$ exists in $0<x<\pi$

$\Rightarrow f\left(x\right)$ is differentiable in $(0,\pi )$

$f\left(0\right)=e^{0}sin0=0,f\left(\pi \right)=e^{\pi }sin\pi =0$

Therefore $f$ satisfies the hypothesis of Rolle's theorem.

Thus there exists $c\in (0,\pi )$ satisfying $f^{\prime }\left(c\right)=0\Rightarrow e^{-c}(cosc-sinc)=0$

$\Rightarrow e^{-c}=0orcosc-sinc=0$

$\Rightarrow e^{-c}=0orcotc=1$

But $e^x$ can never be zero, so $e^{-c}\ne 0$

$\Rightarrow c=\frac{\pi }{4}$

Hence $c=\frac{\pi }{4}$ is the required point.