NCERT Class 12-Mathematics: Chapter – 5 Continuity and Differentiability Part 20 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Verify the Rolle՚s Theorem for Each of the Functions in Exercises 65 to 69

Question 65:



Now, 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:

On expanding , we get

Since, is a polynomial and we know that, every polynomial function is continuous for all

is continuous at

Hence, condition 1 is satisfied.

Condition 2:

Since, f (x) is a polynomial and every polynomial function is differentiable for all x ∈ R

is differentiable at

Hence, condition is satisfied.

Condition 3:


Hence, condition is also satisfied.

Now, let us show that such that

On differentiating above with respect to x, we get

Put in above equation, we get

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

On factorising, we get

So, value of

Thus, Rolle՚s theorem is verified.

Question 66:


We have,

(i) is continuous in

[since, and are continuous functions and we know that, if and be continuous functions, then is a continuous function.]


which exists in

Hence, is differentiable in

(iii) Also,

Conditions of Rolle՚s theorem are satisfied.

Hence, there exists at least one such that


Hence, Rolle՚s theorem has been verified.

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