Question

Give three examples for Rolle's Theorem:  For the first, define f : [0, 1] $\rightarrow$ R such that condition 1 does not hold, condition 2 does hold, condition 3 does hold, and f'(c)$\neq$0 for every c $\in$ (0,1).  For the second example, make sure only condition 2 does not hold and the conclusion do not hold. For the third example, do the same with condition 3.

Answer 1

'f bor Role's Theore m Canditons on conti nuous is Con) 26 is diterentiable 6Co)6C) Qonclusion 3 c 6n ce and such that st Exampl Deline Co IR b 2x+ o<xz 2x+2 Then Also isdisontinuons at x= teventiable at x o ay e Twus kis also nt dik And clearty t Thus is not continuons on is not citerentiabu on (0,1) and 6)o xe() t0

2nd Examblu Detine Co R by -x2 (I')' Then Cleatiscm tinuous on But is not di ferentiabu at x Also Co because cf Corner point ce Co) t(c) o 3rd Exambl Deline Then tis Lntinuans an (0,), is ditteventiabli on 1=t()キ()=0, and 6lerto teeo