SCalcET$94\mathrm{.}2\mathrm{.}013.$

Consider the following function.

f$($x$)=1$ x$2/3$

Find

f$(1)$ and f$(1).$

f$(1)$

$=$

f$(1)$

$=$

Find all values c in $(1,1)$ such that

f$\text{'}($c$)=0.$

$($Enter your answers as a comma$-$separated list. If an answer does not exist, enter DNE.$)$

c $=$

Based off of this information, what conclusions can be made about Rolle's Theorem?

This contradicts Rolle's Theorem, since f is differentiable, f$(1)=$ f$(1),$ and f$\text{'}($c$)=0$ exists, but c is not in $(1,1).$This does not contradict Rolle's Theorem, since f$\text{'}(0)=0,$ and $0$ is in the interval $(1,1).$This contradicts Rolle's Theorem, since f$(1)=$ f$(1),$ there should exist a number c in $(1,1)$ such that f$\text{'}($c$)=0.$This does not contradict Rolle's Theorem, since f$\text{'}(0)$ does not exist, and so f is not differentiable on $(1,1).$Nothing can be concluded.