Exercise 5.8
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NCERT Solutions class 12 Continuity & Differentiability
1. Verify Rolle’s theorem for 
Ans. Consider 
(i) Function is continuous in 
as it is a polynomial function and polynomial function is always continuous.
(ii) 
exists in , hence derivable.
(iii) 
and 
Conditions of Rolle’s theorem are satisfied, hence there exists, at least one 
such that 
NCERT Solutions class 12 Maths Exercise 5.8
2. Examine if Rolles/ theorem is applicable to any of the following functions. Can you say something about the converse of Rolle’s theorem from these examples:
(i) 
for 
(ii) for 
(iii) 
for 
Ans. (i) Being greatest integer function the given function is not differentiable and continuous
hence Rolle’s theorem is not applicable.
(ii) Being greatest integer function the given function is not differentiable and continuous hence Rolle’s theorem is not applicable.
(iii) 
Hence, Rolle’s theorem is not applicable.
NCERT Solutions class 12 Maths Exercise 5.8
3. If 
R is a differentiable function and if does not vanish anywhere, then prove that 
Ans. For, Rolle’s theorem, if
(i) 
is continuous is 
(ii) is derivable in
(iii)
Then, 
It is given that is continuous and derivable, but 
4. Verify Mean Value Theorem if 
in the interval where 
and 
Ans. (i) Function is continuous in [1, 4] as it is a polynomial function and polynomial function is always continuous.
(ii) 
exists in [1, 4], hence derivable. Conditions of MVT theorem are satisfied, hence there exists, at least one 
such that
NCERT Solutions class 12 Maths Exercise 5.8
5. Verify Mean Value Theorem if 
in the interval where and 
Find all 
for which 
Ans. (i) Function is continuous in [1, 3] as it is a polynomial function and polynomial function is always continuous.
(ii) 
exists in [1, 3], hence derivable. Conditions of MVT theorem are satisfied, hence there exists, at least one such that
or 
or 
or
and other value
Since 
, therefore the value of 
does not exist such that 
.
NCERT Solutions class 12 Maths Exercise 5.8
6. Examine the applicability of Mean Value Theorem for all the three functions being given below:
(i) for
(ii) for
(iii) for
Ans. Mean Value Theorem states that for a function 
R, if
(i) is continuous on 
(ii) is differentiable on
Then there exist some 
such that 
Therefore, the Mean Value Theorem is not applicable to those functions that do not satisfy any of the two conditions of the hypothesis.
(i) for
It is evident that the given function 
is not continuous at 
and 
Therefore,
is not continuous at 
Now let 
be an integer such that 
L.H.L. = 
And R.H.L. = 
Since, L.H.L. 
R.H.L.,
Therefore is not differentiable at
Hence Mean Value Theorem is not applicable for for
(ii) for
It is evident that the given function is not continuous at 
and 
Therefore,
is not continuous at 
Now let be an integer such that 
L.H.L. =
And R.H.L. =
Since, L.H.L. R.H.L.,
Therefore is not differentiable at
Hence Mean Value Theorem is not applicable for for
(iii) for ……….(i)
Here, is a polynomial function of degree 2.
Therefore, is continuous and derivable everywhere i.e., on the real time 
Hence is continuous in the closed interval [1, 2] and derivable in open interval (1, 2).
Therefore, both conditions of Mean Value Theorem are satisfied.
Now, From eq. (i), 
Again, From eq. (i), 
And From eq. (ii), 
Therefore, Mean Value Theorem is verified.