Question

Check whether Rolle's theorem is applicable for $f\left(x\right)=2+{(x-1)}^{3/2}$ in the interval $[0,2]$

Answer 1

Given, $f\left(x\right)=2+{(x-1)}^{3/2}=2+(x-1)\sqrt{(x-1)}$
$\Rightarrow f\left(0\right)=2+(0-1)\sqrt{-1}$, not a real number,
Hence rolles theorem is not applicable for this function.