RD Sharma Class 12 Solutions - Differentiability | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


10. Differentiability
Exercise 10.1
1. Question
Show that f(x) = |x - 3| is continuous but not differentiable at x = 3.
Answer
f(x) = |x – 3|
Therefore we can write it as,
f(3) = 3 - 3 = 0
LHL = 
= 
= 
= 
RHL = 
= 
= 
= 0
LHL = RHL = f(3)
Since, f(x) is continuous at x = 3
(LHD at x = 3) = 
= 
= 
= 
= - 1
(RHD at x = 3) = 
= 
= 
= 
= 1
(LHD at x = 3) (RHD at x = 3)
Page 2


10. Differentiability
Exercise 10.1
1. Question
Show that f(x) = |x - 3| is continuous but not differentiable at x = 3.
Answer
f(x) = |x – 3|
Therefore we can write it as,
f(3) = 3 - 3 = 0
LHL = 
= 
= 
= 
RHL = 
= 
= 
= 0
LHL = RHL = f(3)
Since, f(x) is continuous at x = 3
(LHD at x = 3) = 
= 
= 
= 
= - 1
(RHD at x = 3) = 
= 
= 
= 
= 1
(LHD at x = 3) (RHD at x = 3)
Hence, f(x) is continuous but not differentiable at x = 3.
2. Question
Show that f(x) =  is not differentiable at x = 0.
Answer
For differentiability,
LHD(at x = 0) = RHD (at x = 0)
(LHD at x = 0) = 
= 
= 
= 
= 
= 
= Not defined
(RHD at x = 3) = 
= 
= 
= 
= 
= Not defined
Since, LHD and RHD does not exist at x = 0
Hence, f(x) is not differentiable at x = 0
3. Question
Show that  is differentiable at x = 3. Also, find f’(3).
Answer
For differentiability,
LHD(at x = 3) = RHD (at x = 3)
(LHD at x = 3) = 
Page 3


10. Differentiability
Exercise 10.1
1. Question
Show that f(x) = |x - 3| is continuous but not differentiable at x = 3.
Answer
f(x) = |x – 3|
Therefore we can write it as,
f(3) = 3 - 3 = 0
LHL = 
= 
= 
= 
RHL = 
= 
= 
= 0
LHL = RHL = f(3)
Since, f(x) is continuous at x = 3
(LHD at x = 3) = 
= 
= 
= 
= - 1
(RHD at x = 3) = 
= 
= 
= 
= 1
(LHD at x = 3) (RHD at x = 3)
Hence, f(x) is continuous but not differentiable at x = 3.
2. Question
Show that f(x) =  is not differentiable at x = 0.
Answer
For differentiability,
LHD(at x = 0) = RHD (at x = 0)
(LHD at x = 0) = 
= 
= 
= 
= 
= 
= Not defined
(RHD at x = 3) = 
= 
= 
= 
= 
= Not defined
Since, LHD and RHD does not exist at x = 0
Hence, f(x) is not differentiable at x = 0
3. Question
Show that  is differentiable at x = 3. Also, find f’(3).
Answer
For differentiability,
LHD(at x = 3) = RHD (at x = 3)
(LHD at x = 3) = 
= 
= 
= 
= 
= 12
(RHD at x = 3) = 
= 
= 
= 
= 
= 
= 12
Since, (LHD at x = 3) = (RHD at x = 3)
Hence, f(x) is differentiable at x = 3.
4. Question
Show that the function f defined as follows,
Is continuous at x = 2, but not differentiable there at x = 2.
Answer
For continuity,
LHl(at x = 2) = RHL (at x = 2)
f(2) = 2(2)
2
 - 2
= 8 - 2 = 6
LHL = 
= 
= 
= 8 - 2
= 6
RHL = 
Page 4


10. Differentiability
Exercise 10.1
1. Question
Show that f(x) = |x - 3| is continuous but not differentiable at x = 3.
Answer
f(x) = |x – 3|
Therefore we can write it as,
f(3) = 3 - 3 = 0
LHL = 
= 
= 
= 
RHL = 
= 
= 
= 0
LHL = RHL = f(3)
Since, f(x) is continuous at x = 3
(LHD at x = 3) = 
= 
= 
= 
= - 1
(RHD at x = 3) = 
= 
= 
= 
= 1
(LHD at x = 3) (RHD at x = 3)
Hence, f(x) is continuous but not differentiable at x = 3.
2. Question
Show that f(x) =  is not differentiable at x = 0.
Answer
For differentiability,
LHD(at x = 0) = RHD (at x = 0)
(LHD at x = 0) = 
= 
= 
= 
= 
= 
= Not defined
(RHD at x = 3) = 
= 
= 
= 
= 
= Not defined
Since, LHD and RHD does not exist at x = 0
Hence, f(x) is not differentiable at x = 0
3. Question
Show that  is differentiable at x = 3. Also, find f’(3).
Answer
For differentiability,
LHD(at x = 3) = RHD (at x = 3)
(LHD at x = 3) = 
= 
= 
= 
= 
= 12
(RHD at x = 3) = 
= 
= 
= 
= 
= 
= 12
Since, (LHD at x = 3) = (RHD at x = 3)
Hence, f(x) is differentiable at x = 3.
4. Question
Show that the function f defined as follows,
Is continuous at x = 2, but not differentiable there at x = 2.
Answer
For continuity,
LHl(at x = 2) = RHL (at x = 2)
f(2) = 2(2)
2
 - 2
= 8 - 2 = 6
LHL = 
= 
= 
= 8 - 2
= 6
RHL = 
= 
= 
= 6
Since, LHL = RHL = f(2)
Hence, F(x) is continuous at x = 2
For differentiability,
LHD(at x = 2) = RHD (at x = 2)
(LHD at x = 2) = 
= 
= 
= 
= 
= 
= 
= 6
(RHD at x = 2) = 
= 
= 
= 
= 5
Since, (RHD at x = 2) (LHD at x = 2)
Hence, f(2) is not differentiable at x = 2.
5. Question
Discuss the continuity and differentiability of f(x) = |x| + |x - 1| in the interval ( - 1,2).
Answer
f(x) = 
f(x) = 
We know that a polynomial and a constant function is continuous and differentiable every where. So, f(x) is
continuous and differentiable for x ( - 1,0) and x (0,1) and (1,2).
Page 5


10. Differentiability
Exercise 10.1
1. Question
Show that f(x) = |x - 3| is continuous but not differentiable at x = 3.
Answer
f(x) = |x – 3|
Therefore we can write it as,
f(3) = 3 - 3 = 0
LHL = 
= 
= 
= 
RHL = 
= 
= 
= 0
LHL = RHL = f(3)
Since, f(x) is continuous at x = 3
(LHD at x = 3) = 
= 
= 
= 
= - 1
(RHD at x = 3) = 
= 
= 
= 
= 1
(LHD at x = 3) (RHD at x = 3)
Hence, f(x) is continuous but not differentiable at x = 3.
2. Question
Show that f(x) =  is not differentiable at x = 0.
Answer
For differentiability,
LHD(at x = 0) = RHD (at x = 0)
(LHD at x = 0) = 
= 
= 
= 
= 
= 
= Not defined
(RHD at x = 3) = 
= 
= 
= 
= 
= Not defined
Since, LHD and RHD does not exist at x = 0
Hence, f(x) is not differentiable at x = 0
3. Question
Show that  is differentiable at x = 3. Also, find f’(3).
Answer
For differentiability,
LHD(at x = 3) = RHD (at x = 3)
(LHD at x = 3) = 
= 
= 
= 
= 
= 12
(RHD at x = 3) = 
= 
= 
= 
= 
= 
= 12
Since, (LHD at x = 3) = (RHD at x = 3)
Hence, f(x) is differentiable at x = 3.
4. Question
Show that the function f defined as follows,
Is continuous at x = 2, but not differentiable there at x = 2.
Answer
For continuity,
LHl(at x = 2) = RHL (at x = 2)
f(2) = 2(2)
2
 - 2
= 8 - 2 = 6
LHL = 
= 
= 
= 8 - 2
= 6
RHL = 
= 
= 
= 6
Since, LHL = RHL = f(2)
Hence, F(x) is continuous at x = 2
For differentiability,
LHD(at x = 2) = RHD (at x = 2)
(LHD at x = 2) = 
= 
= 
= 
= 
= 
= 
= 6
(RHD at x = 2) = 
= 
= 
= 
= 5
Since, (RHD at x = 2) (LHD at x = 2)
Hence, f(2) is not differentiable at x = 2.
5. Question
Discuss the continuity and differentiability of f(x) = |x| + |x - 1| in the interval ( - 1,2).
Answer
f(x) = 
f(x) = 
We know that a polynomial and a constant function is continuous and differentiable every where. So, f(x) is
continuous and differentiable for x ( - 1,0) and x (0,1) and (1,2).
We need to check continuity and differentiability at x = 0 and x = 1.
Continuity at x = 0
 = 1
 = 1
F(0) = 1
Since, f(x) is continuous at x = 0
Continuity at x = 1
 = 1
 = 1
F(1) = 1
 = 1
Since, f(x) is continuous at x = 1
For differentiability,
LHD(at x = 0) = RHD (at x = 0)
Differentiability at x = 0
(LHD at x = 0) = 
= 
= 
= 2
(RHD at x = 0) = 
= 
= 
= 0
Since,(LHD at x = 0) (RHD at x = 0)
So, f(x) is differentiable at x = 0.
For differentiability,
LHD(at x = 1) = RHD (at x = 1)
Differentiability at x = 1
(LHD at x = 1) = 
= 
= 0