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Examine if Rolle's theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle's theorem from these example? $(i)\;f (x) = [x] \;for\; x \: \in [5,9] $

This is first part of multipart q2

1 Answer

  • Let $f:[a,b]\rightarrow R$ be continuous on [a,b] and differentiable on (a,b).Such that $f(a)=f(b)$ where a and b are some real numbers.Then there exists some $c$ in $(a,b)$ such that $f'(c)=0$
$f(x)=[x]$ for $x\in [5,9]$
In the interval $[5,9],f(x)=[x]$ is neither continuous nor derivable at $x=6,7,8$.Hence Rolle's theorem is not applicable.
$f(x)=[x]$ is the greatest integer less than or equal to $x$
$f'(x)=0$.But f is neither continuous nor differentiable in the interval $[5,9]$
How f'(x) is zero
Can you please explain the converse part
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