Answer :

Given:

Now, we have to show that f(x) verify the Rolle’s Theorem


First of all, Conditions of Rolle’s theorem are:


a) f(x) is continuous at (a,b)


b) f(x) is derivable at (a,b)


c) f(a) = f(b)


If all three conditions are satisfied then there exist some ‘c’ in (a,b) such that f’(c) = 0


Condition 1:


Firstly, we have to show that f(x) is continuous.


Here, f(x) is continuous because f(x) has a unique value for each x [-2,2]


Condition 2:


Now, we have to show that f(x) is differentiable




[using chain rule]



f’(x) exists for all x (-2, 2)


So, f(x) is differentiable on (-2,2)


Hence, Condition 2 is satisfied.


Condition 3:



Now, we have to show that f(a) = f(b)


so, f(a) = f(-2)



and f(b) = f(2)



f(-2) = f(2) = 0


Hence, condition 3 is satisfied


Now, let us show that c (0,1) such that f’(c) = 0



On differentiating above with respect to x, we get



Put x = c in above equation, we get



Thus, all the three conditions of Rolle’s theorem is satisfied. Now we have to see that there exist c (-2,2) such that


f’(c) = 0



c = 0


c = 0 (-2, 2)


Hence, Rolle’s theorem is verified


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