Q. 675.0( 1 Vote )

Verify the Rolle’s theorem for each of the functions

f(x) = log (x2 + 2) – log3 in [– 1, 1].

Answer :

Given: f(x) = log (x2 + 2) – log3

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:


f(x) = log (x2 + 2) – log3


Since, f(x) is a logarithmic function and logarithmic function is continuous for all values of x.


f(x) = log (x2 + 2) – log3 is continuous at x [-1,1]


Hence, condition 1 is satisfied.


Condition 2:


f(x) = log (x2 + 2) – log3



On differentiating above with respect to x, we get







f(x) is differentiable at [-1,1]


Hence, condition 2 is satisfied.


Condition 3:





f(-1) = f(1)


Hence, condition 3 is also satisfied.


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




Put x = c in above equation, we get



, all the three conditions of Rolle’s theorem are satisfied


f’(c) = 0



2c = 0


c = 0 (-1, 1)


Thus, Rolle’s theorem is verified.


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