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NCERT Class 12- Mathematics: Chapter – 5 Continuity and Differentiability Part 20
Verify the Rolle՚s Theorem for Each of the Functions in Exercises 65 to 69
Question 65:
Answer:
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:
Answer:
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.