NCERT Class 12-Mathematics: Chapter –5 Continuity and Differentiability Part 21

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Question 67:


Given: w, we have to show that verify the Rolle’s Theorem

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

(a) is continuous at

(b) is derivable at


If all three conditions are satisfied then there exist some ‘’ in such that

Condition 1:

nce, is a logarithmic function and logarithmic function is continuous for all values of .

is continuous at x ∈ [-1,1]

Hence, condition 1 is satisfied.

Condition 2:

On differentiating above with respect to , we get

is differentiable at

Hence, condition is satisfied.

Condition 3:

Hence, condition 3 is also satisfied.

Now, let us show that such that

Put in above equation, we get

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

Thus, Rolle’s theorem is verified.

Question 68:



We have,

[i] is a continuous function. [since, it is a combination of polynomial functions and an exponential function which are continuous functions]

So, is continuous in


Hence, is differentiable in



Since, conditions of Rolle’s theorem are satisfied.

Hence, there exists a real number such that


Therefore, Rolle’s theorem has been verified.