NCERT Solutions Class 12

NCERT Solutions Class 12

Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 8)

 Exercise 5.8

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 

 

 


 

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.

 


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

 

 

 


 

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 .


 

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.

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