Rd Sharma XII Vol 1 2020 Solutions for Class 12 Science Maths Chapter 9 Differentiability are provided here with simple step-by-step explanations. These solutions for Differentiability are extremely popular among Class 12 Science students for Maths Differentiability Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma XII Vol 1 2020 Book of Class 12 Science Maths Chapter 9 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma XII Vol 1 2020 Solutions. All Rd Sharma XII Vol 1 2020 Solutions for class Class 12 Science Maths are prepared by experts and are 100% accurate.
Page No 9.10:
Question 1:
Show that f(x) = |x − 2| is continuous but not differentiable at x = 2.
Answer:
Given:
Continuity at x=2: We have,
(LHL at x = 2)
.
(RHL at x = 2)
.
and
Thus, = = .
Hence, is continuous at .
Differentiability at x = 2: We have,
(LHD at x = 2)
(RHD at x=2)
=
Thus, ≠ .
Hence, is not differentiable at x=2 .
Page No 9.10:
Question 2:
Show that f(x) = x1/3 is not differentiable at x = 0.
Answer:
Disclaimer: It might be a wrong question because f(x) is differentiable at x=0
Given: .
We have,
(LHD at x = 0)
(RHD at x = 0)
LHD at (x = 0)= RHD at (x = 0)
Hence, is differentiable at x = 0
Page No 9.10:
Question 3:
Show that is differentiable at x = 3. Also, find f'(3).
Answer:
Given:
We have to show that the given function is differentiable at x = 3.
We have,
(LHD at x=3) =
(RHD at x = 3) =
Thus, (LHD at x=3) = (RHD at x=3) = 12.
So, is differentiable at x=3 and
Page No 9.10:
Question 4:
Show that the function f defined as follows, is continuous at x = 2, but not differentiable thereat:
Answer:
Given:
=
First , we will show that f(x) is continuos at .
We have,
(LHL at x=2)
(RHL at x = 2)
and
Thus, = = .
Hence the function is continuous at x=2.
Now, we will check whether the given function is differentiable at x = 2.
We have,
(LHD at x = 2)
(RHD at x = 2)
Thus, LHD at x=2 ≠ RHD at x = 2.
Hence, function is not differentiable at x = 2.
Page No 9.10:
Question 5:
Discuss the continuity and differentiability of the
Answer:
Now,
Page No 9.10:
Question 6:
Find whether the function is differentiable at x = 1 and x = 2
Answer:
Page No 9.10:
Question 7:
Show that the function
(i) differentiable at x = 0, if m > 1
(ii) continuous but not differentiable at x = 0, if 0 < m < 1
(iii) neither continuous nor differentiable, if m ≤ 0
Answer:
Given:
x≠0 , x=0
(i) Let m=2, then the function becomes , x≠0, x=0
Differentiability at x=0:
[ ∵ , as (∵ for all ) and hence when ]
So, , which means f is differentiable at x=0.
Hence the given function is differentiable at x=0.
(ii) Let . Then the function becomes
, x≠0 , x=0
Continuity at x=0:
(LHL at x=0) = .
(RHL at x=0) = .
and
LHL at x=0 = RHL at x=0 = ,
Hence continuous.
Now Differentiabilty at x=0 when 0<m<1.
(LHD at x=0) =
Page No 9.10:
Question 8:
Find the values of a and b so that the function is differentiable at each x ∈ R.
Answer:
Given:
It is given that the function is differentiable at each and every differentiable function is continuous.
So, is continuous at .
Therefore,
Since, is differentiable at . So,
(LHD at x = 1) = (RHD at x = 1)
From , we have
Hence, .
Page No 9.10:
Question 9:
Show that the function is continuous but not differentiable at x = 1.
Answer:
Given:
Continuity at x = 1:
(LHL at x = 1) =
(RHL at x = 1) =
Hence, (LHL at x = 1) = (RHL at x = 1)
Differentiability at x = 1:
LHD ≠ RHD
Hence, the function is continuous but not differentiable at x = 1.
Page No 9.11:
Question 10:
If is differentiable at x = 1, find a, b.
Answer:
Given:
It is given that the given function is differentiable at x = 1.
We know every differentiable function is continuous. Therefore it is continuous at x=1. Then,
It is also differentiable at x=1. Therefore,
(LHD at x = 1) = (RHD at x = 1)
From (i), we have:
Hence, when and the function is differentiable at x = 1.
Page No 9.11:
Question 11:
Find the values of a and b, if the function f defined by is differentiable at x = 1.
Answer:
Given that f(x) is differentiable at x = 1. Therefore, f(x) is continuous at x = 1.
Again, f(x) is differentiable at x = 1. So,
(LHD at x = 1) = (RHD at x = 1)
Putting b = 5 in (1), we get
a = 3
Hence, a = 3 and b = 5.
Page No 9.16:
Question 1:
If f is defined by f (x) = x2, find f'(2).
Answer:
Given: .
We know a polynomial function is everywhere differentiable. Therefore is differentiable at .
Page No 9.16:
Question 2:
If f is defined by , show that
Answer:
Given:
Clearly, being a polynomial function, is everywhere differentiable. The derivative of at is given by:
Now,
Therefore,
Hence proved.
Page No 9.16:
Question 3:
Show that the derivative of the function f given by
, at x = 1 and x = 2 are equal.
Answer:
Given:
Clearly, being a polynomial function, is differentiable everywhere. Therefore the derivative of at is given by:
So,
Hence the derivative at and are equal.
Page No 9.16:
Question 4:
If for the function
Answer:
Given:
Clearly, being a polynomial function, is differentiable everywhere. Therefore the derivative of at is given by:
It is given
Thus,
Page No 9.16:
Question 5:
If , find f'(4).
Answer:
Given:
Clearly, being a polynomial function, is differentiable everywhere. Therefore the derivative of at is given by:
Thus,
Page No 9.16:
Question 6:
Find the derivative of the function f defined by f (x) = mx + c at x = 0.
Answer:
Given:
Clearly, being a polynomial function, is differentiable everywhere. Therefore the derivative of at is given by:
Thus,
Page No 9.16:
Question 7:
Examine the differentialibilty of the function f defined by
Answer:
Page No 9.16:
Question 8:
Write an example of a function which is everywhere continuous but fails to differentiable exactly at five points.
Answer:
The above function is continuous everywhere but not differentiable at x = 0, 1, 2, 3 and 4
Page No 9.16:
Question 9:
Discuss the continuity and differentiability of f (x) = |log |x||.
Answer:
We have,
f (x) = |log |x||
Here, LHD ≠ RHD
So, function is not differentiable at x = − 1
At 0 function is not defined.
Here, LHD ≠ RHD
So, function is not differentiable at x = 1
Hence, function is not differentiable at x = 0 and ± 1
At 0 function is not defined.
So, at 0 function is not continuous.
Hence, function f (x) = |log |x|| is not continuous at x = 0
Page No 9.16:
Question 10:
Discuss the continuity and differentiability of f (x) = e|x| .
Answer:
Given:
Continuity:
(LHL at x = 0)
(RHL at x = 0)
and
Thus,
Hence,function is continuous at x = 0 .
Differentiability at x = 0.
(LHD at x = 0)
(RHD at x = 0)
LHD at (x = 0)RHD at (x = 0)
Hence the function is not differentiable at x = 0.
Page No 9.16:
Question 11:
Discuss the continuity and differentiability of
Answer:
Given:
Continuity:
(LHL at x = c)
(RHL at x = c)
and
Differentiability at x = c
(LHD at x = c)
Page No 9.16:
Question 12:
Is |sin x| differentiable? What about cos |x|?
Answer:
Let, f(x) = |sin x|
Page No 9.17:
Question 5:
Let . Then, for all x
(a) f is continuous
(b) f is differentiable for some x
(c) f' is continuous
(d) f'' is continuous
Answer:
(a) f is continuous
(c) f' is continuous
Page No 9.17:
Question 6:
The function f (x) = e−|x| is
(a) continuous everywhere but not differentiable at x = 0
(b) continuous and differentiable everywhere
(c) not continuous at x = 0
(d) none of these
Answer:
(a) continuous everywhere but not differentiable at x = 0
Given:
RHL at x = 0
and f(0) =
Thus,
Hence, function is continuous at x = 0
Differentiability at x = 0
(LHD at x = 0)
Therefore, left hand derivative does not exist.
Hence, the function is not differentiable at x = 0.
Page No 9.17:
Question 7:
The function f (x) = |cos x| is
(a) everywhere continuous and differentiable
(b) everywhere continuous but not differentiable at (2n + 1) π/2, n ∈ Z
(c) neither continuous nor differentiable at (2n + 1) π/2, n ∈ Z
(d) none of these
Answer:
(b) everywhere continuous but not differentiable at (2n + 1) π/2, n ∈ Z
Page No 9.17:
Question 8:
If
(a) continuous on [−1, 1] and differentiable on (−1, 1)
(b) continuous on [−1, 1] and differentiable on
(c) continuous and differentiable on [−1, 1]
(d) none of these
Answer:
Page No 9.17:
Question 9:
If and if f (x) is differentiable at x = 0, then
(a)
(b)
(c)
(d)
Answer:
(b)
Page No 9.17:
Question 1:
Let f (x) = |x| and g (x) = |x3|, then
(a) f (x) and g (x) both are continuous at x = 0
(b) f (x) and g (x) both are differentiable at x = 0
(c) f (x) is differentiable but g (x) is not differentiable at x = 0
(d) f (x) and g (x) both are not differentiable at x = 0
Answer:
Option (a) f (x) and g (x) both are continuous at x = 0
Given:
We know is continuous at x=0 but not differentiable at x = 0 as (LHD at x = 0) ≠ (RHD at x = 0).
Now, for the function
Continuity at x = 0:
(LHL at x = 0) =
(RHL at x = 0) =
and
Thus, .
Hence, is continuous at x = 0.
Differentiability at x = 0:
(LHD at x = 0) =
(RHD at x = 0) =
Thus, (LHD at x = 0) = (RHD at x = 0).
Hence, the function is differentiable at x = 0.
Page No 9.17:
Question 2:
The function f (x) = sin−1 (cos x) is
(a) discontinuous at x = 0
(b) continuous at x = 0
(c) differentiable at x = 0
(d) none of these
Answer:
(b) continuous at x = 0
Given:
Continuity at x = 0:
We have,
(LHL at x = 0)
(RHL at x = 0)
Differentiability at x = 0:
(LHD at x = 0)
RHD at x = 0
Hence, the function is not differentiable at x = 0 but is continuous at x = 0.
Page No 9.17:
Question 3:
The set of points where the function f (x) = x |x| is differentiable is
(a)
(b)
(c)
(d)
Answer:
(a)
Page No 9.17:
Question 4:
If , then f (x) is
(a) continuous at x = − 2
(b) not continuous at x = − 2
(c) differentiable at x = − 2
(d) continuous but not derivable at x = − 2
Answer:
(b) not continuous at x = − 2
Given:
Continuity at x = − 2.
(LHL at x= − 2) =
(RHL at x = −2) =
Also
Thus, ≠
Therefore, given function is not continuous at x = − 2
Page No 9.18:
Question 10:
If
then at x = 0, f (x)
(a) has no limit
(b) is discontinuous
(c) is continuous but not differentiable
(d) is differentiable
Answer:
(b) is discontinuous
Page No 9.18:
Question 11:
If
(a)
(b)
(c)
(d)
Answer:
(a) and (b)
Page No 9.18:
Question 12:
If , then
(a) f (x) is continuous and differentiable for all x in its domain
(b) f (x) is continuous for all for all × in its domain but not differentiable at x = ± 1
(c) f (x) is neither continuous nor differentiable at x = ± 1
(d) none of these
Answer:
(b) f (x) is continuous for all x in its domain but not differentiable at x = ± 1
And we know that logarithmic function is continuous in its domain.
Therefore, given function is continuous for all x in its domain but not differentiable at x = ± 1
Page No 9.18:
Question 13:
Let
If f (x) is continuous and differentiable at any point, then
(a)
(b)
(c) a = 1, b = − 1
(d) none of these
Answer:
(b)
Page No 9.18:
Question 14:
The function f (x) = x − [x], where [⋅] denotes the greatest integer function is
(a) continuous everywhere
(b) continuous at integer points only
(c) continuous at non-integer points only
(d) differentiable everywhere
Answer:
(c) continuous at non-integer points only
Therefore, given points are continuous at non-integer points only.
Page No 9.18:
Question 15:
Let . Then, f (x) is derivable at x = 1, if
(a) a = 2
(b) a = 1
(c) a = 0
(d) a = 1/2
Answer:
(d) a = 1/2
Given:
The function is derivable at x = 1, iff left hand derivative and right hand derivative of the function are equal at x = 1.
Page No 9.18:
Question 16:
Let f (x) = |sin x|. Then,
(a) f (x) is everywhere differentiable.
(b) f (x) is everywhere continuous but not differentiable at x = n π, n ∈ Z
(c) f (x) is everywhere continuous but not differentiable at .
(d) none of these
Answer:
(b) f (x) is everywhere continuous but not differentiable at x = n π, n ∈ Z
Page No 9.18:
Question 17:
Let f (x) = |cos x|. Then,
(a) f (x) is everywhere differentiable
(b) f (x) is everywhere continuous but not differentiable at x = n π, n ∈ Z
(c) f (x) is everywhere continuous but not differentiable at .
(d) none of these
Answer:
(c) f (x) is everywhere continuous but not differentiable at .
Page No 9.18:
Question 18:
The function f (x) = 1 + |cos x| is
(a) continuous no where
(b) continuous everywhere
(c) not differentiable at x = 0
(d) not differentiable at x = n π, n ∈ Z
Answer:
(b) continuous everywhere
Graph of the function f (x) = 1 + |cos x| is as shown below:
From the graph, we can see that f (x) is everywhere continuous but not differentiable at
Page No 9.18:
Question 19:
The function f (x) = |cos x| is
(a) differentiable at x = (2n + 1) π/2, n ∈ Z
(b) continuous but not differentiable at x = (2n + 1) π/2, n ∈ Z
(c) neither differentiable nor continuous at x = n ∈ Z
(d) none of these
Answer:
(b) continuous but not differentiable at x = (2n + 1) π/2, n ∈ Z
Page No 9.18:
Question 20:
The function , where [⋅] denotes the greatest integer function, is
(a) continuous as well as differentiable for all x ∈ R
(b) continuous for all x but not differentiable at some x
(c) differentiable for all x but not continuous at some x.
(d) none of these
Answer:
(a) continuous as well as differentiable for all x ∈ R
Here,
Since, we know that and .
∵
∴f(x) = 0 for all x
Thus, f(x) is a constant function and it is continuous and differentiable everywhere.
Page No 9.19:
Question 21:
Let f (x) = a + b |x| + c |x|4, where a, b, and c are real constants. Then, f (x) is differentiable at x = 0, if
(a) a = 0
(b) b = 0
(c) c = 0
(d) none of these
Answer:
(b) b = 0
Page No 9.19:
Question 22:
If f (x) = |3 − x| + (3 + x), where (x) denotes the least integer greater than or equal to x, then f (x) is
(a) continuous and differentiable at x = 3
(b) continuous but not differentiable at x = 3
(c) differentiable nut not continuous at x = 3
(d) neither differentiable nor continuous at x = 3
Answer:
(d) neither differentiable nor continuous at x = 3
Page No 9.19:
Question 23:
If then f (x) is
(a) continuous as well as differentiable at x = 0
(b) continuous but not differentiable at x = 0
(c) differentiable but not continuous at x = 0
(d) none of these
Answer:
(d) none of these
we have,
So, f(x) is not continuous at x = 0
Differentiability at x = 0
Page No 9.19:
Question 24:
If
then at x = 0, f (x) is
(a) continuous and differentiable
(b) differentiable but not continuous
(c) continuous but not differentiable
(d) neither continuous nor differentiable
Answer:
(a) continuous and differentiable
we have,
Hence, f(x)is continuous at x = 0.
For differentiability at x = 0
Page No 9.19:
Question 25:
The set of points where the function f (x) given by f (x) = |x − 3| cos x is differentiable, is
(a) R
(b) R − {3}
(c) (0, ∞)
(d) none of these
Answer:
(b)
So, f(x) is not differentiable at x = 3.
Also, f(x) is differentiable at all other points because both modulus and cosine functions are differentiable and the product of two differentiable function is differentiable.
Page No 9.19:
Question 26:
Let Then, f is
(a) continuous at x = − 1
(b) differentiable at x = − 1
(c) everywhere continuous
(d) everywhere differentiable
Answer:
(b) differentiable at x = − 1
Differentiabilty at x = − 1
(LHD x = − 1)
(RHD x = − 1)
Page No 9.19:
Question 27:
The function f(x) = e|x| is
(a) continuous every where but not differentiable at x = 0
(b) continuous and differentiable everywhere
(c) not continuous at x = 0
(d) none of these
Answer:
The given function is f(x) = e|x|.
We know
If f is continuous on its domain D, then is also continuous on D.
Now, the identity function p(x) = x is continuous everywhere.
So, g(x) = is also continuous everywhere.
Also, the exponential function ax, a > 0 is continuous everywhere.
So, h(x) = ex is continuous everywhere.
The composition of two continuous functions is continuous everywhere.
is continuous everywhere.
Now,
And
So, is not differentiable at x = 0.
We know
The exponential function ax, a > 0 is differentiable everywhere.
So, h(x) = ex is differentiable everywhere.
We know that, the composition of differentiable functions is differentiable.
Now, ex is differentiable everywhere, but is not differentiable at x = 0.
is differentiable everywhere except at x = 0.
Thus, the function f(x) = e|x| is continuous every where but not differentiable at x = 0.
Hence, the correct answer is option (a).
Page No 9.19:
Question 28:
The set of points where the function f(x) = |2x – 1| sin x is differentiable, is
(a) R
(b)
(c) (0, ∞)
(d) none of these
Answer:
Let and .
We know that, the trigonometric functions are differentiable in their respective domain.
So, is differentiable for all x ∈ R.
Now,
(2x − 1) and −(2x − 1) are polynomial functions which are differentiable at each x ∈ R. So, f(x) is differentiable for all and for all .
So, we need to check the differentiability of g(x) at .
We have
And
So, is not differentiable at .
The function is differentiable for all .
We know that, the product of two differentiable functions is differentiable.
is differentiable for all .
Thus, the set of points where the function is differentiable is .
Hence, the correct answer is option (b).
Page No 9.20:
Question 1:
The function f(x) = |x + 1| is not differentiable at x = ____________.
Answer:
The given function is .
Now, (x + 1) and −(x + 1) are polynomial functions which are differentiable at each x ∈ R. So, f(x) is differentiable for all and for all .
So, we need to check the differentiability of f(x) at .
We have,
And
So, f(x) is not differentiable at .
Thus, the function f(x) = |x + 1| is not differentiable at x = −1.
The function f(x) = |x + 1| is not differentiable at x = ___−1___.
Page No 9.20:
Question 2:
The function g(x) = |x – 1| + |x + 1| is not differentiable at x = ____________.
Answer:
When x < −1, g(x) = −2x which being a polynomial function is continuous and differentiable.
When −1 ≤ x < 1, g(x) = 2 which being a constant function is continuous and differentiable.
When x ≥ 1, g(x) = 2x which being a polynomial function is continuous and differentiable.
Thus, the possible points of non-differentiability of g(x) are x = −1 and x = 1.
Now,
And
So, g(x) is not differentiable at x = −1.
Also,
And
So, g(x) is not differentiable at x = 1.
Thus, the function is not differentiable at x = −1 and x = 1.
The function g(x) = |x – 1| + |x + 1| is not differentiable at x = ___±1___.
Page No 9.20:
Question 3:
The set of points where f(x) = x – [x] not differentiable is ____________.
Answer:
Let g(x) = x and h(x) = [x].
Every polynomial function is differentiable for all x ∈ R. So, g(x) = x is differentiable for all x ∈ R.
Also, the function h(x) = [x] is discontinuous at all integral values of x i.e. x ∈ Z. So, h(x) = [x] is not differentiable at all integral values of x i.e. x ∈ Z.
Now, f(x) = g(x) − h(x) = x − [x]
So, the function f(x) = x − [x] is differentiable for all x ∈ R except at all integral values of x i.e. x ∈ Z. The function f(x) = x − [x] is not differentiable for all x ∈ R − Z.
Thus, the set of points where f(x) = x – [x] not differentiable is R − Z.
The set of points where f(x) = x – [x] not differentiable is ___R − Z___.
Page No 9.20:
Question 4:
The number of points in [–π, π] where f(x) = sin–1 (sin x) is not differentiable is. ____________.
Answer:
Let us check the differentiability of the function at and .
At ,
So, the function f(x) is not differentiable at .
At ,
So, the function f(x) is not differentiable at .
Thus, the function f(x) = sin–1(sinx), x ∈ [–, ] is not differentiable at and .
The number of points in [–π, π] where f(x) = sin–1 (sin x) is not differentiable is .
Page No 9.20:
Question 5:
The function f(x) = cos–1(cos x), x ∈ (–2π, 2π) is not differentiable at x = ____________.
Answer:
Let us check the differentiability of the function at , x = 0 and .
At ,
So, the function f(x) is not differentiable at .
At ,
So, the function f(x) is not differentiable at .
At ,
So, the function f(x) is not differentiable at .
Thus, the function f(x) = cos–1(cos x), x ∈ (–2, 2) is not differentiable at , x = 0 and .
The function f(x) = cos–1(cos x), x ∈ (–2, 2) is not differentiable at x = .
Page No 9.20:
Question 6:
The function is not differentiable at x = ____________.
Answer:
We know that, is not differentiable at x = 0.
Therefore, is not differentiable when .
, n ∈ Z
Now, the only value of x lying in given interval at which the function is not differentiable is 0.
Thus, the function is not differentiable at x = 0.
The function is not differentiable at x = ___0___.
Page No 9.20:
Question 7:
Let If f(x) is differentiable at x = 1, then a = ____________.
Answer:
The given function is differentiable at x = 1.
Here, LHS is finite.
So, for RHS to be finite, we must have
Thus, the value of a is .
Let If f(x) is differentiable at x = 1, then a = .
Page No 9.20:
Question 8:
If f(x) = x |x|, then f' (–1) = ____________.
Answer:
Now,
Also,
So,
If f(x) = x|x|, then f'(–1) = ___2___.
Page No 9.20:
Question 9:
If f(x) = x |x|, then f' (2) =____________.
Answer:
Now,
Also,
So, .
If f(x) = x|x|, then f'(2) = ___4____.
Page No 9.20:
Question 10:
The set of point where the function f(x) = |2x – 1| is differentiable, is ____________.
Answer:
The given function is .
Now, (2x − 1) and −(2x − 1) are polynomial functions which are differentiable at each x ∈ R. So, f(x) is differentiable for all and for all .
So, we need to check the differentiability of f(x) at .
We have,
And
So, f(x) is not differentiable at .
Thus, the set of points where the function f(x) = |2x – 1| is differentiable is .
The set of point where the function f(x) = |2x – 1| is differentiable, is .
Page No 9.20:
Question 11:
The set of points where the function is not differentiable, is ____________.
Answer:
The given function is .
(x + 1) and (2x − 1) are polynomial functions which are differentiable at each x ∈ R. So, f(x) is differentiable for all x < 2 and for all x > 2.
So, we need to check the differentiability of f(x) at x = 2.
We have
And
So, f(x) is not differentiable at x = 2.
Thus, the set of points where the function f(x) is not differentiable is {2}.
The set of points where the function is not differentiable, is _____{2}______.
Page No 9.20:
Question 12:
An example of a function which is everywhere continuous but fails to be differentiable exactly at two points is ____________.
Answer:
Consider the function .
When x < −1, g(x) = −2x which being a polynomial function is continuous and differentiable.
When −1 ≤ x < 1, g(x) = 2 which being a constant function is continuous and differentiable.
When x ≥ 1, g(x) = 2x which being a polynomial function is continuous and differentiable.
Let us check the continuity and differentiability of g(x) at x = −1 and x = 1.
At x = −1,
LHL =
RHL =
Since , so the function g(x) is continuous at x = −1.
At x = 1,
LHL =
RHL =
Since , so the function g(x) is continuous at x = 1.
Thus, the function g(x) is continuous everywhere i.e. for all x ∈ R.
Now,
And
So, g(x) is not differentiable at x = −1.
Also,
And
So, g(x) is not differentiable at x = 1.
Thus, the function g(x) is differentiable everywhere except at x = −1 and x = 1.
An example of a function which is everywhere continuous but fails to be differentiable exactly at two points is .
Page No 9.20:
Question 13:
The set of points where f(x) = cos |x| is differentiable, is ____________.
Answer:
We know
We know that, cosine function is differentiable in its domain. So, f(x) is differentiable for all x < 0 and x > 0.
Let us check the differentiability of at x = 0.
Now,
And
So, f(x) is differentiable at x = 0. Thus, the function f(x) is differentiable everywhere.
Hence, the set of points where is differentiable is R (set of real real numbers).
The set of points where f(x) = cos |x| is differentiable, is _____R_____.
Page No 9.20:
Question 14:
The set of points where f(x) = |sin x| is not differentiable, is ____________.
Answer:
Let
Now,
And
So, is not differentiable at x = 0.
Therefore, is not differentiable when .
Thus, the set of points where is not differentiable is .
The set of points where f(x) = |sin x| is not differentiable, is .
Page No 9.20:
Question 15:
The set of points at which the function is not differentiable, is ____________.
Answer:
The given function is .
For f(x) to be defined,
and
and
and
Thus, the function f(x) is not defined when x = −1, x = 0 and x = 1.
We know that, the logarithmic function is differentiable at each point in its domain. Every constant function is differentiable at each x ∈ R. Also, the quotient of two differentiable functions is differentiable.
So, the function is not differentiable at x = −1, x = 0 and x = 1.
Thus, the set of points at which the function is not differentiable is {−1, 0, 1}.
The set of points at which the function is not differentiable, is ___{−1, 0, 1}___.
Page No 9.20:
Question 1:
Define differentiability of a function at a point.
Answer:
Let be a real valued function defined on an open interval and let .
Then is said to be differentiable or derivable at iff
exists finitely.
or,
Page No 9.20:
Question 2:
Is every differentiable function continuous?
Answer:
Yes, if a function is differentiable at a point then it is necessary continuous at that point.
Page No 9.20:
Question 3:
Is every continuous function differentiable?
Answer:
No, function may be continuous at a point but may not be differentiable at that point .
For example: function is continuous at but it is not differentiable at .
Page No 9.20:
Question 4:
Give an example of a function which is continuos but not differentiable at at a point.
Answer:
Consider a function,
This mod function is continuous at x=0 but not differentiable at x=0.
Continuity at x=0, we have:
(LHL at x = 0)
(RHL at x = 0)
and
Thus,
Hence, is continuous at
Now, we will check the differentiability at x=0, we have:
(LHD at x = 0)
(RHD at x = 0)
Thus, ≠
Hence is not differentiable at .
Page No 9.20:
Question 5:
If f (x) is differentiable at x = c, then write the value of .
Answer:
Given: is differentiable at . Then,
exists finitely.
or, .
Consider,
Page No 9.20:
Question 6:
If f (x) = |x − 2| write whether f' (2) exists or not.
Answer:
Given:
Now,
(LHD at x = 2)
(RHD at x = 2)
Thus, (LHD at x = 2) ≠ (RHD at x = 2)
Hence, does not exist.
Page No 9.20:
Question 7:
Write the points where f (x) = |loge x| is not differentiable.
Answer:
Given:
Clearly is differentiable for all and for all . So, we have to check the differentiability at .
We have,
(LHD at x = 1)
(RHD at x=1)
=
Thus, (LHD at x =1) ≠ (RHD at x =1)
So, is not differentiable at
Page No 9.20:
Question 8:
Write the points of non-differentiability of
Answer:
We have,
f (x) = |log |x||
Here, LHD ≠ RHD
So, function is not differentiable at x = − 1
At 0 function is not defined.
Here, LHD ≠ RHD
So, function is not differentiable at x = 1
Hence, function is not differentiable at x = 0 and ± 1
Page No 9.20:
Question 9:
Write the derivative of f (x) = |x|3 at x = 0.
Answer:
Given:
(LHD at x = 0)
.
(RHD at x = 0)
and
Thus, (LHD at x=0) = (RHD at x = 0) =
Hence,
Page No 9.20:
Question 10:
Write the number of points where f (x) = |x| + |x − 1| is continuous but not differentiable.
Answer:
Given:
When , we have:
which, being a polynomial function is continuous and differentiable.
When , we have:
which, being a constant function is continuous and differentiable on (0,1).
When , we have:
which, being a polynomial function is continuous and differentiable on .
Thus, the possible points of non- differentiability of are 0 and 1.
Now,
(LHD at x = 0)
[∵ ]
(RHD at x = 0)
=
[∵ ]
Thus, (LHD at x=0) ≠ (RHD at x=0)
Hence is not differentiable at
Now, is not differentiable at .
(LHD at x = 1)
(RHD at x = 1)
=
Thus, (LHD at x =1) ≠ (RHD at x=1)
.
Hence is not differentiable at .
Therefore, 0,1 are the points where f(x) is continuous but not differentiable.
Page No 9.21:
Question 11:
If exists finitely, write the value of .
Answer:
Given: exists finitely. Then,
.
Now,
Page No 9.21:
Question 12:
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
Answer:
Given:
We check differentiability at x = 2
(LHD at x = 2)
Page No 9.21:
Question 13:
If , write the value of
Answer:
Given:
Now,
So,
On rationalising the numerator, we get
Taking limit , we have
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