RD Sharma XII Vol 1 2017 Solutions for Class 12 Science Math Chapter 10 Differentiability are provided here with simple step-by-step explanations. These solutions for Differentiability are extremely popular among class 12 Science students for Math Differentiability Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the RD Sharma XII Vol 1 2017 Book of class 12 Science Math Chapter 10 are provided here for you for free. You will also love the ad-free experience on Meritnation’s RD Sharma XII Vol 1 2017 Solutions. All RD Sharma XII Vol 1 2017 Solutions for class 12 Science Math are prepared by experts and are 100% accurate.

#### Question 1:

Show that f(x) = |x − 2| is continuous but not differentiable at x = 2.

Given:

Continuity at x=2: We have,

(LHL at x = 2)
.

(RHL at x = 2)
.

and

Thus,  =  = $f\left(2\right)$$f\left(2\right)$.
Hence, $f\left(x\right)$ is continuous at $x=2$.

Differentiability at x = 2: We have,

(LHD at x = 2)

(RHD at x=2)

Thus, ≠ .

Hence, $f\left(x\right)$ is not differentiable at x=2 .

#### Question 2:

Show that f(x) = x1/3 is not differentiable at x = 0.

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

#### Question 3:

Show that is differentiable at x = 3. Also, find f'(3).

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, $f\left(x\right)$ is differentiable at x=3 and

#### Question 4:

Show that the function f defined as follows, is continuous at x = 2, but not differentiable thereat: $f\left(x\right)=\left\{\begin{array}{ll}3x-2,& 02\end{array}\right\$

Given:
$f\left(x\right)$ =

First , we will show that f(x) is continuos at $x=2$.

We have,

(LHL at x=2)

(RHL at x = 2)

and

Thus,   =  = $f\left(2\right)$.

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.

#### Question 5:

Discuss the continuity and differentiability of the

Now,

#### Question 6:

Find whether the function is differentiable at x = 1 and x = 2
$f\left(x\right)=\left\{\begin{array}{ll}x& x\le 1\\ \begin{array}{c}2-x\\ -2+3x-{x}^{2}\end{array}& \begin{array}{c}1\le x\le 2\\ x>2\end{array}\end{array}\right\$

$f\left(x\right)=\left\{\begin{array}{ll}x& x\le 1\\ \begin{array}{c}2-x\\ -2+3x-{x}^{2}\end{array}& \begin{array}{c}1\le x\le 2\\ x>2\end{array}\end{array}\right\\phantom{\rule{0ex}{0ex}}⇒f\text{'}\left(x\right)=\left\{\begin{array}{ll}1& x\le 1\\ \begin{array}{c}-1\\ 3-2x\end{array}& \begin{array}{c}1\le x\le 2\\ x>2\end{array}\end{array}\right\\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

#### 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

Given:
x≠0 , x=0

(i) Let m=2, then the function becomes  ,  x≠0, x=0

Differentiability at x=0:

[ ∵ , as (∵ for all $\theta$) and hence  when $\left|x-0\right|<\epsilon$$\left|x-0\right|<\epsilon$ ]
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) =

#### Question 8:

Find the values of a and b so that the function is differentiable at each xR.

Given:

It is given that the function is differentiable at each $x\in R$ and every differentiable function is continuous.
So, $f\left(x\right)$ is continuous at $x=1$.

Therefore,

Since, $f\left(x\right)$ is differentiable at $x=1$. So,

(LHD at x = 1) = (RHD at x = 1)

From $\left(i\right)$, we have

Hence, .

#### Question 9:

Show that the function is continuous but not differentiable at x = 1.

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.

#### Question 10:

If is differentiable at x = 1, find a, b.

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 $a=-\frac{1}{2}$ and $b=-\frac{3}{2}$ the function is differentiable at x = 1.

#### Question 1:

If f is defined by f (x) = x2, find f'(2).

Given: .

We know a  polynomial function is everywhere differentiable. Therefore $f\left(x\right)$ is differentiable at $x=2$.

#### Question 2:

If f is defined by $f\left(x\right)={x}^{2}-4x+7$, show that $f\text{'}\left(5\right)=2f\text{'}\left(\frac{7}{2}\right)$

Given:

Clearly, $f\left(x\right)$ being a polynomial function, is everywhere differentiable. The derivative of $f$ at $x$ is given by:

Now,

Therefore,
Hence proved.

#### Question 3:

Show that the derivative of the function f given by
$f\left(x\right)=2{x}^{3}-9{x}^{2}+12x+9$, at x = 1 and x = 2 are equal.

Given:

Clearly, being a polynomial function, is differentiable everywhere. Therefore the derivative of $f$ at $x$ is given by:

So,

Hence the derivative at $x=1$ and $x=2$ are equal.

#### Question 4:

If for the function

Given:

Clearly, being a polynomial function, is differentiable everywhere. Therefore the derivative of $\varphi$ at $x$ is given by:

It is given
Thus,

#### Question 5:

If $f\left(x\right)={x}^{3}+7{x}^{2}+8x-9$, find f'(4).

Given:

Clearly, being a polynomial function, is differentiable everywhere. Therefore the derivative of $f$ at $x$ is given by:

Thus,

#### Question 6:

Find the derivative of the function f defined by f (x) = mx + c at x = 0.

Given:

Clearly, being a polynomial function, is differentiable everywhere. Therefore the derivative of $f$ at $x$ is given by:

Thus,

#### Question 7:

Examine the differentialibilty of the function f defined by

#### Question 8:

Write an example of a function which is everywhere continuous but fails to differentiable exactly at five points.

$f\left(x\right)=\left|x\right|+\left|x+1\right|+\left|x+2\right|+\left|x+3\right|+\left|x+4\right|$
The above function is continuous everywhere but not differentiable at x = 0, 1, 2, 3 and 4

#### Question 9:

Discuss the continuity and differentiability of f (x) = |log |x||.

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

#### Question 10:

Discuss the continuity and differentiability of f (x) = e|x| .

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)$\ne$RHD at (x = 0)

Hence the function is not differentiable at x = 0.

#### Question 11:

Discuss the continuity and differentiability of

Given:

Continuity:

(LHL at x = c)

(RHL at x = c

and
Differentiability at x = c

(LHD at x = c)

#### Question 12:

Is |sin x| differentiable? What about cos |x|?

Let, f(x) = |sin x|

#### Question 1:

Define differentiability of a function at a point.

Let $f\left(x\right)$ be a real valued function defined on an open interval $\left(a,b\right)$ and let $c\in \left(a,b\right)$.
Then $f\left(x\right)$ is said to be differentiable or derivable at $x=c$ iff
exists finitely.

or,

#### Question 2:

Is every differentiable function continuous?

Yes, if a function is differentiable at a point then it is necessary continuous at that point.

#### Question 3:

Is every continuous function differentiable?

No, function may be continuous at a point but may not be differentiable at that point .
For example: function   is continuous at $x=0$ but it is not differentiable at $x=0$.

#### Question 4:

Give an example of a function which is continuos but not differentiable at at a point.

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, $f\left(x\right)$ is continuous at $x=0.$

Now, we will check the differentiability at x=0, we have:

(LHD at x = 0)

(RHD at x = 0)

Thus,   ≠

Hence $f\left(x\right)$ is not differentiable at $x=0$.

#### Question 5:

If f (x) is differentiable at x = c, then write the value of .

Given: $f\left(x\right)$ is differentiable at $x=c$. Then,
exists finitely.

or,  .

Consider,

#### Question 6:

If f (x) = |x − 2| write whether f' (2) exists or not.

Given:

Now,

(LHD at x = 2)

(RHD at x = 2)

Thus, (LHD at x = 2) ≠ (RHD at x = 2)

Hence,  does not exist.

#### Question 7:

Write the points where f (x) = |loge x| is not differentiable.

Given:
Clearly $f\left(x\right)$ is differentiable for all $x>1$ and for all $x<1$. So, we have to check the differentiability at $x=1$.
We have,
(LHD at x = 1)

(RHD at x=1)

Thus, (LHD at x =1) ≠ (RHD at x =1)
So, $f\left(x\right)$ is not differentiable at $x=1.$

#### Question 8:

Write the points of non-differentiability of

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

#### Question 9:

Write the derivative of f (x) = |x|3 at x = 0.

Given:

(LHD at x = 0)
.

(RHD at x = 0)

and

Thus, (LHD at x=0) = (RHD at x = 0) = $f\left(0\right)$

Hence,

#### Question 10:

Write the number of points where f (x) = |x| + |x − 1| is continuous but not differentiable.

Given:

When $x<0$, we have:

which, being a polynomial function is continuous and differentiable.

When $0\le x<1$, we have:
which, being a constant function is continuous and differentiable on (0,1).

When $x\ge 1$, we have:
which, being a polynomial function is continuous and differentiable on $x>2$.

Thus, the possible points of non- differentiability of $f\left(x\right)$ 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 $x=0$

Now, $f\left(x\right)$ is not differentiable at $x=1$.

(LHD at x = 1)

(RHD at x = 1)
=

Thus, (LHD at x =1) ≠ (RHD at x=1)
.
Hence $f\left(x\right)$ is not differentiable at $x=1$.

Therefore, 0,1 are the points where f(x) is continuous but not differentiable.

#### Question 11:

If $\underset{x\to c}{\mathrm{lim}}\frac{f\left(x\right)-f\left(c\right)}{x-c}$ exists finitely, write the value of $\underset{x\to c}{\mathrm{lim}}f\left(x\right)$.

Given:   exists finitely. Then,

.

Now,

#### Question 12:

Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.

Given:

We check differentiability at x = 2

(LHD at x = 2)

#### Question 13:

If , write the value of

Given:
Now,

So,

On rationalising the numerator, we get

Taking limit $x\to 4$, we have

#### 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

Option (a) f (x) and g (x) both are continuous at x = 0

Given:

We know  $\left|x\right|$ 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, $g\left(x\right)$ 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  $g\left(x\right)$ is differentiable at x = 0.

#### 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

(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.

#### Question 3:

The set of points where the function f (x) = x |x| is differentiable is
(a)
(b)
(c)
(d)

(a)

#### 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

(b) not continuous at x = − 2

Given:

Continuity at x = − 2.
(LHL at x= − 2) =

(RHL at x = −2) =

Also
Thus,  ≠ $f\left(-2\right).$
Therefore, given function is not continuous at x = − 2

#### 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

(a) f is continuous
(c) f' is continuous

#### 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

(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.

#### 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, nZ
(c) neither continuous nor differentiable at (2n + 1) π/2, nZ
(d) none of these

(b) everywhere continuous but not differentiable at (2n + 1) π/2, nZ

#### 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

#### Question 9:

If and if f (x) is differentiable at x = 0, then
(a) $a=b=c=0$
(b)
(c)
(d)

(b)

#### Question 10:

If $f\left(x\right)={x}^{2}+\frac{{x}^{2}}{1+{x}^{2}}+\frac{{x}^{2}}{\left(1+{x}^{2}\right)}+...+\frac{{x}^{2}}{\left(1+{x}^{2}\right)}+....,$

then at x = 0, f (x)
(a) has no limit
(b) is discontinuous
(c) is continuous but not differentiable
(d) is differentiable

(b) is discontinuous

If
(a)
(b)
(c)
(d)

(a) and (b)

#### 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

(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

#### 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

(b)

#### 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

(c) continuous at non-integer points only

Therefore, given points are continuous at non-integer points only.

#### Question 15:

Let $f\left(x\right)\left\{\begin{array}{ll}a{x}^{2}+1,& x>1\\ x+1/2,& x\le 1\end{array}\right\$. Then, f (x) is derivable at x = 1, if

(a) a = 2
(b) a = 1
(c) a = 0
(d) a = 1/2

(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.

#### 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, nZ
(c) f (x) is everywhere continuous but not differentiable at .
(d) none of these

(b) f (x) is everywhere continuous but not differentiable at x = n π, nZ

#### 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, nZ
(c) f (x) is everywhere continuous but not differentiable at .
(d) none of these

(c) f (x) is everywhere continuous but not differentiable at .

#### 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, nZ

(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

#### Question 19:

The function f (x) =  |cos x| is
(a) differentiable at x = (2n + 1) π/2, nZ
(b) continuous but not differentiable at x = (2n + 1) π/2, nZ
(c) neither differentiable nor continuous at x = nZ
(d) none of these

(b) continuous but not differentiable at x = (2n + 1) π/2, nZ

#### 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

(a) continuous as well as differentiable for all x ∈ R

Here,

Since, we know that $\mathrm{\pi }\left[\left(x-\mathrm{\pi }\right)\right]=n\mathrm{\pi }$ and $\mathrm{sin}n\mathrm{\pi }=0$.

$4+{\left[x\right]}^{2}\ne 0$

f(x) = 0 for all x

Thus, f(x) is a constant function and it is continuous and differentiable everywhere.

#### 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

(b) b = 0

#### 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

(d) neither differentiable nor continuous at x = 3

#### 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

(d) none of these

we have,

So, f(x) is not continuous at x = 0

Differentiability at x = 0

#### 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

(a) continuous and differentiable

we have,

Hence, f(x)is continuous at x = 0.

For differentiability at x = 0

#### 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

(b) $R-\left(3\right)$

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.

#### Question 26:

Let Then, f is

(a) continuous at x = − 1
(b) differentiable at x = − 1
(c) everywhere continuous
(d) everywhere differentiable

(b) differentiable at x = − 1

Differentiabilty at x = − 1
(LHD x = − 1)

(RHD x = − 1)

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