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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?

 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]


 

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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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