RD Sharma Class 12 Solutions - Mean Value Theorems

 Page 1


15. Mean Value Theorems
Exercise 15.1
1 A. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x) = 3 + (x – 2)
2/3
 on [1, 3]
Answer
First, let us write the conditions for the applicability of Rolle’s theorem:
For a Real valued function ‘f’:
a) The function ‘f’ needs to be continuous in the closed interval [a,b].
b) The function ‘f’ needs differentiable on the open interval (a,b).
c) f(a) = f(b)
Then there exists at least one c in the open interval (a,b) such that f’(c) = 0.
(i) Given function is:
?  on [1,3]
Let us check the differentiability of the function f(x).
Let’s find the derivative of f(x),
? 
? 
? 
? 
? 
Let’s the differentiability at the value of x = 2
? 
? 
? 
? 
? f is not differentiable at x = 2, so it is not differentiable in the closed interval (1,3).
So, Rolle’s theorem is not applicable for the function f on the interval [1,3].
1 B. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x = [x] for – 1 = x = 1, where [x] denotes the greatest integer not exceeding x
Page 2


15. Mean Value Theorems
Exercise 15.1
1 A. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x) = 3 + (x – 2)
2/3
 on [1, 3]
Answer
First, let us write the conditions for the applicability of Rolle’s theorem:
For a Real valued function ‘f’:
a) The function ‘f’ needs to be continuous in the closed interval [a,b].
b) The function ‘f’ needs differentiable on the open interval (a,b).
c) f(a) = f(b)
Then there exists at least one c in the open interval (a,b) such that f’(c) = 0.
(i) Given function is:
?  on [1,3]
Let us check the differentiability of the function f(x).
Let’s find the derivative of f(x),
? 
? 
? 
? 
? 
Let’s the differentiability at the value of x = 2
? 
? 
? 
? 
? f is not differentiable at x = 2, so it is not differentiable in the closed interval (1,3).
So, Rolle’s theorem is not applicable for the function f on the interval [1,3].
1 B. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x = [x] for – 1 = x = 1, where [x] denotes the greatest integer not exceeding x
Answer
Given function is:
? f(x) = [x], – 1=x=1 where [x] denotes the greatest integer not exceeding x.
Let us check the continuity of the function ‘f’.
Here in the interval x?[ – 1,1], the function has to be Right continuous at x = 1 and left continuous at x = 1.
? 
?  where h>0.
? 
?  ......(1)
? 
? , where h>0
? 
?  ......(2)
From (1) and (2), we can see that the limits are not the same so, the function is not continuous in the interval
[ – 1,1].
? Rolle’s theorem is not applicable for the function f in the interval [ – 1,1].
1 C. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x) = sin 1/x for – 1 = x = 1
Answer
Given function is:
?  for – 1=x=1
Let us check the continuity of the function ‘f’ at the value of x = 0.
We can not directly find the value of limit at x = 0, as the function is not valid at x = 0. So, we take the limit
on either sides and x = 0, and we check whether they are equal or not.
Right – Hand Limit:
? 
We assume that the limit , k?[ – 1,1].
? , where h>0
? 
? 
?  ...... (1)
Left – Hand Limit:
Page 3


15. Mean Value Theorems
Exercise 15.1
1 A. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x) = 3 + (x – 2)
2/3
 on [1, 3]
Answer
First, let us write the conditions for the applicability of Rolle’s theorem:
For a Real valued function ‘f’:
a) The function ‘f’ needs to be continuous in the closed interval [a,b].
b) The function ‘f’ needs differentiable on the open interval (a,b).
c) f(a) = f(b)
Then there exists at least one c in the open interval (a,b) such that f’(c) = 0.
(i) Given function is:
?  on [1,3]
Let us check the differentiability of the function f(x).
Let’s find the derivative of f(x),
? 
? 
? 
? 
? 
Let’s the differentiability at the value of x = 2
? 
? 
? 
? 
? f is not differentiable at x = 2, so it is not differentiable in the closed interval (1,3).
So, Rolle’s theorem is not applicable for the function f on the interval [1,3].
1 B. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x = [x] for – 1 = x = 1, where [x] denotes the greatest integer not exceeding x
Answer
Given function is:
? f(x) = [x], – 1=x=1 where [x] denotes the greatest integer not exceeding x.
Let us check the continuity of the function ‘f’.
Here in the interval x?[ – 1,1], the function has to be Right continuous at x = 1 and left continuous at x = 1.
? 
?  where h>0.
? 
?  ......(1)
? 
? , where h>0
? 
?  ......(2)
From (1) and (2), we can see that the limits are not the same so, the function is not continuous in the interval
[ – 1,1].
? Rolle’s theorem is not applicable for the function f in the interval [ – 1,1].
1 C. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x) = sin 1/x for – 1 = x = 1
Answer
Given function is:
?  for – 1=x=1
Let us check the continuity of the function ‘f’ at the value of x = 0.
We can not directly find the value of limit at x = 0, as the function is not valid at x = 0. So, we take the limit
on either sides and x = 0, and we check whether they are equal or not.
Right – Hand Limit:
? 
We assume that the limit , k?[ – 1,1].
? , where h>0
? 
? 
?  ...... (1)
Left – Hand Limit:
? 
? , where h>0
? 
? 
? 
? 
?  ......(2)
From (1) and (2), we can see that the Right hand and left – hand limits are not equal, so the function ‘f’ is not
continuous at x = 0.
? Rolle’s theorem is not applicable to the function ‘f’ in the interval [ – 1,1].
1 D. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x) = 2x
2
 – 5x + 3 on [1, 3]
Answer
Given function is:
? f(x) = 2x
2
 – 5x + 3 on [1,3]
Since given function ‘f’ is a polynomial. So, it is continuous and differentiable every where.
Now, we find the values of function at the extremum values.
? f(1) = 2(1)
2
–5(1) + 3
? f(1) = 2 – 5 + 3
? f(1) = 0 ......(1)
? f(3) = 2(3)
2
–5(3) + 3
? f(3) = 2(9)–15 + 3
? f(3) = 18 – 12
? f(3) = 6 ......(2)
From (1) and (2), we can say that,
f(1)?f(3)
? Rolle’s theorem is not applicable for the function f in interval [1,3].
1 E. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x) = x
2/3
 on [ – 1, 1]
Answer
Given function is:
?  on [ – 1,1]
Page 4


15. Mean Value Theorems
Exercise 15.1
1 A. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x) = 3 + (x – 2)
2/3
 on [1, 3]
Answer
First, let us write the conditions for the applicability of Rolle’s theorem:
For a Real valued function ‘f’:
a) The function ‘f’ needs to be continuous in the closed interval [a,b].
b) The function ‘f’ needs differentiable on the open interval (a,b).
c) f(a) = f(b)
Then there exists at least one c in the open interval (a,b) such that f’(c) = 0.
(i) Given function is:
?  on [1,3]
Let us check the differentiability of the function f(x).
Let’s find the derivative of f(x),
? 
? 
? 
? 
? 
Let’s the differentiability at the value of x = 2
? 
? 
? 
? 
? f is not differentiable at x = 2, so it is not differentiable in the closed interval (1,3).
So, Rolle’s theorem is not applicable for the function f on the interval [1,3].
1 B. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x = [x] for – 1 = x = 1, where [x] denotes the greatest integer not exceeding x
Answer
Given function is:
? f(x) = [x], – 1=x=1 where [x] denotes the greatest integer not exceeding x.
Let us check the continuity of the function ‘f’.
Here in the interval x?[ – 1,1], the function has to be Right continuous at x = 1 and left continuous at x = 1.
? 
?  where h>0.
? 
?  ......(1)
? 
? , where h>0
? 
?  ......(2)
From (1) and (2), we can see that the limits are not the same so, the function is not continuous in the interval
[ – 1,1].
? Rolle’s theorem is not applicable for the function f in the interval [ – 1,1].
1 C. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x) = sin 1/x for – 1 = x = 1
Answer
Given function is:
?  for – 1=x=1
Let us check the continuity of the function ‘f’ at the value of x = 0.
We can not directly find the value of limit at x = 0, as the function is not valid at x = 0. So, we take the limit
on either sides and x = 0, and we check whether they are equal or not.
Right – Hand Limit:
? 
We assume that the limit , k?[ – 1,1].
? , where h>0
? 
? 
?  ...... (1)
Left – Hand Limit:
? 
? , where h>0
? 
? 
? 
? 
?  ......(2)
From (1) and (2), we can see that the Right hand and left – hand limits are not equal, so the function ‘f’ is not
continuous at x = 0.
? Rolle’s theorem is not applicable to the function ‘f’ in the interval [ – 1,1].
1 D. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x) = 2x
2
 – 5x + 3 on [1, 3]
Answer
Given function is:
? f(x) = 2x
2
 – 5x + 3 on [1,3]
Since given function ‘f’ is a polynomial. So, it is continuous and differentiable every where.
Now, we find the values of function at the extremum values.
? f(1) = 2(1)
2
–5(1) + 3
? f(1) = 2 – 5 + 3
? f(1) = 0 ......(1)
? f(3) = 2(3)
2
–5(3) + 3
? f(3) = 2(9)–15 + 3
? f(3) = 18 – 12
? f(3) = 6 ......(2)
From (1) and (2), we can say that,
f(1)?f(3)
? Rolle’s theorem is not applicable for the function f in interval [1,3].
1 E. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x) = x
2/3
 on [ – 1, 1]
Answer
Given function is:
?  on [ – 1,1]
Let’s find the derivative of the given function:
? 
? 
? 
? 
Let’s check the differentiability of the function at x = 0.
? 
? 
? 
Since the limit for the derivative is undefined at x = 0, we can say that f is not differentiable at x = 0.
? Rolle’s theorem is not applicable to the function ‘f’ on [ – 1,1].
1 F. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
Answer
Given function is:
? 
Let’s check the continuity at x = 1 as the equation of function changes.
Left – Hand Limit:
? 
? 
?  ......(1)
Right – Hand Limit:
? 
? 
?  ......(2)
From (1) and (2), we can see that the values of both side limits are not equal. So, the function ‘f’ is not
continuous at x = 1.
? Rolle’s theorem is not applicable to the function ‘f’ in the interval [0,2].
2 A. Question
Page 5


15. Mean Value Theorems
Exercise 15.1
1 A. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x) = 3 + (x – 2)
2/3
 on [1, 3]
Answer
First, let us write the conditions for the applicability of Rolle’s theorem:
For a Real valued function ‘f’:
a) The function ‘f’ needs to be continuous in the closed interval [a,b].
b) The function ‘f’ needs differentiable on the open interval (a,b).
c) f(a) = f(b)
Then there exists at least one c in the open interval (a,b) such that f’(c) = 0.
(i) Given function is:
?  on [1,3]
Let us check the differentiability of the function f(x).
Let’s find the derivative of f(x),
? 
? 
? 
? 
? 
Let’s the differentiability at the value of x = 2
? 
? 
? 
? 
? f is not differentiable at x = 2, so it is not differentiable in the closed interval (1,3).
So, Rolle’s theorem is not applicable for the function f on the interval [1,3].
1 B. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x = [x] for – 1 = x = 1, where [x] denotes the greatest integer not exceeding x
Answer
Given function is:
? f(x) = [x], – 1=x=1 where [x] denotes the greatest integer not exceeding x.
Let us check the continuity of the function ‘f’.
Here in the interval x?[ – 1,1], the function has to be Right continuous at x = 1 and left continuous at x = 1.
? 
?  where h>0.
? 
?  ......(1)
? 
? , where h>0
? 
?  ......(2)
From (1) and (2), we can see that the limits are not the same so, the function is not continuous in the interval
[ – 1,1].
? Rolle’s theorem is not applicable for the function f in the interval [ – 1,1].
1 C. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x) = sin 1/x for – 1 = x = 1
Answer
Given function is:
?  for – 1=x=1
Let us check the continuity of the function ‘f’ at the value of x = 0.
We can not directly find the value of limit at x = 0, as the function is not valid at x = 0. So, we take the limit
on either sides and x = 0, and we check whether they are equal or not.
Right – Hand Limit:
? 
We assume that the limit , k?[ – 1,1].
? , where h>0
? 
? 
?  ...... (1)
Left – Hand Limit:
? 
? , where h>0
? 
? 
? 
? 
?  ......(2)
From (1) and (2), we can see that the Right hand and left – hand limits are not equal, so the function ‘f’ is not
continuous at x = 0.
? Rolle’s theorem is not applicable to the function ‘f’ in the interval [ – 1,1].
1 D. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x) = 2x
2
 – 5x + 3 on [1, 3]
Answer
Given function is:
? f(x) = 2x
2
 – 5x + 3 on [1,3]
Since given function ‘f’ is a polynomial. So, it is continuous and differentiable every where.
Now, we find the values of function at the extremum values.
? f(1) = 2(1)
2
–5(1) + 3
? f(1) = 2 – 5 + 3
? f(1) = 0 ......(1)
? f(3) = 2(3)
2
–5(3) + 3
? f(3) = 2(9)–15 + 3
? f(3) = 18 – 12
? f(3) = 6 ......(2)
From (1) and (2), we can say that,
f(1)?f(3)
? Rolle’s theorem is not applicable for the function f in interval [1,3].
1 E. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x) = x
2/3
 on [ – 1, 1]
Answer
Given function is:
?  on [ – 1,1]
Let’s find the derivative of the given function:
? 
? 
? 
? 
Let’s check the differentiability of the function at x = 0.
? 
? 
? 
Since the limit for the derivative is undefined at x = 0, we can say that f is not differentiable at x = 0.
? Rolle’s theorem is not applicable to the function ‘f’ on [ – 1,1].
1 F. Question
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
Answer
Given function is:
? 
Let’s check the continuity at x = 1 as the equation of function changes.
Left – Hand Limit:
? 
? 
?  ......(1)
Right – Hand Limit:
? 
? 
?  ......(2)
From (1) and (2), we can see that the values of both side limits are not equal. So, the function ‘f’ is not
continuous at x = 1.
? Rolle’s theorem is not applicable to the function ‘f’ in the interval [0,2].
2 A. Question
Verify Rolle’s theorem for each of the following functions on the indicated intervals :
f(x) = x
2
 – 8x + 12 on [2, 6]
Answer
First let us write the conditions for the applicability of Rolle’s theorem:
For a Real valued function ‘f’:
a) The function ‘f’ needs to be continuous in the closed interval [a,b].
b) The function ‘f’ needs differentiable on the open interval (a,b).
c) f(a) = f(b)
Then there exists at least one c in the open interval (a,b) such that f’(c) = 0.
Given function is:
? f(x) = x
2
 – 8x + 12 on [2,6]
Since, given function f is a polynomial it is continuous and differentiable everywhere i.e., on R.
Let us find the values at extremums:
? f(2) = 2
2
 – 8(2) + 12
? f(2) = 4 – 16 + 12
? f(2) = 0
? f(6) = 6
2
 – 8(6) + 12
? f(6) = 36 – 48 + 12
? f(6) = 0
? f(2) = f(6), Rolle’s theorem applicable for function ‘f’ on [2,6].
Let’s find the derivative of f(x):
? 
? 
? 
? f’(x) = 2x – 8
We have f’(c) = 0 c?(2,6), from the definition given above.
? f’(c) = 0
? 2c – 8 = 0
? 2c = 8
? 
? C = 4?(2,6)
? Rolle’s theorem is verified.
2 B. Question
Verify Rolle’s theorem for each of the following functions on the indicated intervals :
f(x) = x
2
 – 4x + 3 on [1, 3]