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

Get unlimited access to the best preparation resource for CBSE/Class-12 : get questions, notes, tests, video lectures and more- for all subjects of CBSE/Class-12.

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

Question 65:

Given:

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

(c)

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,

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.]

(ii)

which exists in

Hence, is differentiable in

(iii) Also,

Conditions of Rolle՚s theorem are satisfied.

Hence, there exists at least one such that

And

Hence, Rolle՚s theorem has been verified.

Developed by: