Question

Check whether the conditions of Rolle's theorem are satisfied by the following
function or not:

i. $f\left(x\right)=\left(x-1\right)\left(x-2\right)\left(x-3\right),x\in \left[1,3\right]$

ii. $f\left(x\right)=e^{-x}\sin x,xϵ[0,\pi ]$

Answer 1

Here, $f\left(x\right)$ is a cubic equation.

• So, $f\left(x\right)$ is continuous in [1,3].

• And, $f\left(x\right)$ is also differentiable in (1,3)

• => $f\left(1\right)=0=f\left(3\right)$

Hence, rolle's theorem satisfied.