NCERT Solutions for Class 12 Maths Exercise 5.8 Chapter 5 Continuity and Differentiability – FREE PDF Download
Free PDF download of NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.8 (Ex 5.8) and all chapter exercises at one place prepared by expert teacher as per NCERT (CBSE) books guidelines. Class 12 Maths Chapter 5 Continuity and Differentiability Exercise 5.8 Questions with Solutions to help you to revise complete Syllabus and Score More marks.
NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Exercise 5.8 (Ex 5.8)
1. Verify Rolle’s theorem for 
(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 
2. Examine if Rolle’s 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 
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.
3. If 
R is a differentiable function and if does not vanish anywhere, then prove that
R is a differentiable function and if 
(i) 
is continuous in 
(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
in the interval 
and 
(ii) 
exists in [1, 4], hence derivable. Conditions of MVT theorem are satisfied, hence there exists, at least one 
such that
5. Verify Mean Value Theorem if in the interval where and Find all for which
in the interval 
Find all 
(ii) f′(x)=3x2−10x−3f′(x)=3×2−10x−3, it exists in [1, 3], hence derivable.
Conditions of MVT theorem are satisfied, hence there exists, at least one 
such that
−27−(−7)2=3c2−10c−3−27−(−7)2=3c2−10c−3
or 
or 
or
and other value
Since 
, therefore the value of 
does not exist such that 
.
6. Examine the applicability of Mean Value Theorem for all the three functions being given below:
(i) for
(ii) for
(iii) for
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.
