Question

$f\left(x\right)=x^2$ in $2\le x\le 3$ Is Rolle's theorem applicable?

Answer 1

$f\left(x\right)=x^2$ in $2\le x\le 3$
$f^{\prime }\left(x\right)=2x$
$\therefore \underset{h\to 0}{lim}{f}^{\prime }\left(x\right)=\underset{h\to 0}{lim}2x=0.$

Therefore, $I{I}^{nd}$ condition of Rolle's theorem is satisfied.
Also $f\left(2\right)=4,f\left(3\right)=9$
$f\left(2\right)\ne f\left(3\right)\Rightarrow$
Third condition of Rolle's theorem is not satisfied. $f\left(2\right)=x^2$ is continuous and differentiable in $2\le x\le 3.$
Thus the first two conditions of Rolle's theorem are satisfied and third condition is not satisfied. Hence, Rolle's theorem is not applicable.