# Q2 (3)   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?                $f (x) = x^2 - 1 \: \:for \: \: x \epsilon [ 1,2]$

Answers (1)

According to Rolle's theorem function must be
a )  continuous in given closed interval say [x,y]
b ) differentiable in given open interval say (x,y)
c ) f(x) = f(y)
Then there exist a $c \ \epsilon \ (x,y)$  such that   $f^{'}(c)= 0$
If all these conditions are satisfied then we can verify  Rolle's theorem
Given function is
$f (x) = x^2-1$
Now, being a polynomial , function $f (x) = x^2-1$ is continuous in [1,2] and differentiable in(1,2)
Now,
$f(1)=1^2-1 = 1-1 = 0$
And
$f(2)=2^2-1 = 4-1 = 3$
Therefore,  $f(1)\neq f(2)$
Therefore, All conditions are not satisfied
Hence, Rolle's theorem is not applicable for given function   $f (x) = [x]$    ,  $x \ \epsilon \ [-2,2]$

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