RBSE Solutions for Class 12 Maths Chapter 6 Continuity and Differentiability Ex 6.2

Rajasthan Board RBSE Class 12 Maths Chapter 6 Continuity and Differentiability Ex 6.2

Question 1.
Show that following functions are differentiable for every value of x :
(i) Identity function, f (x) = x
(ii) Constant function,f (x) = c, where c is a constant
(iii) f(x) = e^x
(iv) f(x) = sin x.
Solution:
(i) Given, f (x) = x, (identity function)
where, x ∈ R
Let a be arbitrary constant, then
At x = a, Left hand derivative of f (x)


So, for every x, identity function f(x) is differentiable.

(ii) Given, constant function f(x) = c, where c is constant. Domain of function f(x) is set of real numbers (R).
Let a be any arbitrary real number, then
At x = a, Left hand derivative of f (x)

So, for every x, identity function f(x) is differentiable.

(iii) Given function f (x) = e^x, where x ∈ R
Let a be an arbitrary constant then at x = a,
Left hand derivative of f (x)

Again, at x = a, Right hand derivative of f (x)

Hence,f(x) = e^x is differentiable for every x.

(iv) Given function f(x) = sin x, where x ∈ R
Let a be any arbitrary real number.
At x = a, Left hand derivative of (x)

Again, at x = a, Right hand derivative of f (x)


Hence, for every x, function will be differentiable.

Question 2.
Show that the function f (x) = | x | is not differentiable at x = 0.
Solution:
For differentiability at x = 0.

Left hand derivative

Right hand derivative

Hence, f(x) is not differentiable at x = 0.

Question 3.
Examine for differentiability of function,
f (x) = | x – 1 | + | x | at x = 0, 1.
Solution:
Given function can be written as

For differentiability at x = 0,

Left hand derivative

Right hand derivative

So, function f(x) is not differentiable at x = 0
For differentiability at x = 1

Left hand derivative

Right hand derivative


So, function f(x) is not differentiable at x = 1.
Hence, function f(x) is not differentiable at x = 0 and x = 1.

Question 4.
Examine the function for differentiability in interval [0, 2], if
f(x) = | x – 1 | + | x – 2 |
Solution:
Given function can be written as

Here, we will test the differentiability at x = 1.
Since 1 ∈ [0, 2]
For differentiability at x = 1

Left hand derivative

Right hand derivative

Hence, f (x) is not differentiable at x = 1. So, it is not differentiable in interval [0,2].

Question 5.
Examine the function for differentiability at

Solution:
For differentiability at x = 0,

Left hand derivative

Right hand derivative

Hence, function is differentiable at x = 0.

Question 6.
Examine the function f(x) for differentiability at x = 0, if

Solution:
For differentiability at x = 0,

Left hand derivative

Right hand derivative

Hence, function is not differentiable at x = 0.

Question 7.
Show that the following function

(a) Continuous at x = 0 if m > 0
(b) Differentiable at x = 0 if m > 1.
Solution:
(a) Continuity at x = 0,
(i) At x = 0,f (0) = 0
(ii) At x = 0,
Left hand limit

(iii) At x = 0,
Right hand limit


At x = 0, function f(x) will be continuous if (i) and (ii) will be zero.
Since, -1 < cos (\(\frac { 1 }{ h } \)) < 1
Therefore, both limits will be zero if m > 0
Hence, function f(x) will be continue at x = 0 if m > 0.

(b) Differentiability at x = 0
Left hand derivative

Right hand derivative

Given, f(x) is differentiable at x = 0, then
f'(0 – 0) = f'(0 + 0)
Which is possible only when
m – 1 > 0 or m > 1
Hence, given function f(x) is differentiable at x = 0 if m >1.

Question 8.
Examine the function f(x) for differentiability at x = 0 if

Solution:
At x = 0,
Left hand derivative

Right hand derivative

Hence, function is not differentiable at x = 0.

Question 9.
Examine the function f(x) for differentiability

Solution:
At x = 0,
Left hand derivative

Right hand derivative

Hence, given function is not differentiable at x = 0.

Question 10.
Examine the function f(x) for differentiability at x = \(\frac { \pi }{ 2 } \), if

Solution:
At x = \(\frac { \pi }{ 2 } \),

Left hand derivative

Right hand derivative

Question 11.
Find the value of m and n, if function

is differentiable at every point.
Solution:
Given that at x = 1, f(x) is differentiable. We know that every differentiable function is continuous. So, at x = 1, function is continuous also.

Left hand limit

Right hand limit

Left hand derivative

Right hand derivative

∴ Function is differentiable at x = 1
then f'(1 – 0) = f'(1 + 0)
5 = n ⇒ n = 5
From equation (i),
m – 5 = – 2
⇒ m = – 2 + 5
⇒ m = 3
Hence, m = 3 and n = 5

RBSE Solutions for Class 12 Maths