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Q)

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? $(iii)\;f (x) = x^2 - 1 \; for\; x \: \in [1,2] $

This is third part of multipart q2

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A)
Toolbox:
  • 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 $f'(c)=0$
Step 1:
$f(x)=(x^2-1)$
$f(1)=1-1=0$
$f(2)=2^2-1=3$
$f(a)\neq f(b)$
Though it is continuous and derivable in the interval $[1,-2]$ and Rolle's theorem is not applicable.
Step 2:
$f(x)=x^2-1$
$f'(x)=2x$
$f'(x)$ is not zero in the interval $[1,2]$.
So $f(2)\neq f'(2)$
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