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RD Sharma Class 12 Solutions Chapter 16 Exercise 16.2 Increasing and Decreasing Functions consist of formulae and concepts on tracking the homogeneous arrangement of natural conditions. The RD Sharma Class 12th Exercise 16.2 which is titled Increasing and Decreasing Functions will help students to practice these concepts promptly and solve all complex problems. RD Sharma Solutions are considered to be essential and the best reference book that students have used to score high in their board tests.

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- Chapter 16 - Increasing and Decreasing Functions- Ex 16.1
- Chapter 16 - Increasing and Decreasing Function- Ex VSA
- Chapter 16 - Increasing and Decreasing Function- Ex MCQ

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion i

Answer:Given:

Here given that

To find:

We have to find out the intervals in which function is increasing and decreasing.

Hint:

Solution:

Here given that

On differentiating we get,

By applying (-) ve sign change comparison sign.

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion ii

Answer:Given:

Here given that

To find:

We have to find out the intervals in which function is increasing and decreasing.

Hint:

Solution:

Here given that

On differentiating we get,

For f(x) to be increasing, we must have

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion iii

Answer:Given:

Here given that

To find:

We have to find out the intervals in which function is increasing and decreasing.

Hint:

Use increasing and decreasing property.

Solution:

Here given that

On differentiating we get,

For f(x) to be increasing, we must have

By applying (-) ve sign change comparison sign.

For f(x) to be decreasing, we must have

By applying (-) ve sign change comparison sign.

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion iv

Answer:Given:

Here given that

To find:

We have to find out the intervals in which function is increasing and decreasing.

Hint:

Firstly we will find critical points and then use increasing and decreasing property.

Solution:

Given that

On differentiating we get,

Firstly we will find critical points for f(x).

For this we have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion v

Answer:Given:

Here given that

To find:

We have to find out the intervals in which function is increasing and decreasing.

Hint:

First, we will find critical points and then use increasing and decreasing property.

Solution:

Given that

On differentiating we get,

For f(x), Firstly we will find critical points.

For this we have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion vi

Answer:Given:

Here given that

To find:

We have to find out the intervals in which function is increasing and decreasing.

Hint:

First, we will find critical points and then use increasing and decreasing property.

Solution:

Given that

On differentiating we get,

For f(x), Firstly we will find critical points.

For this we have,

I

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion vii

Answer:Given:

Here given that

To find:

We have to find out the intervals in which function is increasing and decreasing.

Hint:

Put f(x)=0 and solve this equation to find critical points of given function.

Solution:

We have,

Differentiating with respect to x we get,

Now we have to find critical points for f(x).

We must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion viii

Answer:Given:

Here given that.

To find:

We have to find the intervals in which function is increasing and decreasing.

Hint:

Put f(x)=0 and solve this equation to find critical points of f(x) and use increasing and decreasing property.

Solution:

We have,

Differentiating with respect to x we get,

Now we have to find critical points for f(x).

We must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion ix

Answer:Given:

Here given that

To find:

We have to find the intervals in which function is increasing and decreasing.

Hint:

Put f ‘(x) = 0 to find critical points of f(x) and use increasing and decreasing property.

Solution:

We have,

Differentiating w.r.t. x, we get,

Now for critical points of f(x), we must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion x

Answer:Given:

Here given that

To find:

We have to find out the intervals in which function is increasing and decreasing.

Hint:

First, we will find critical points and then use increasing and decreasing property.

Solution:

We have,

Differentiating w.r.t. x we get,

For f(x), we have to find critical points.

For this we must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xi

Answer:Given:

Here given that

To find:

We have to find the intervals in which function is increasing and decreasing interval of f(x).

Hint:

Put f ‘(x) = 0 and solve this equation to find critical points of f(x).

Solution:

We have,

Differentiating w.r.t. x, we get,

For f(x) we have to find critical points, for this we must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xii

Answer:Given:

Here given that

To find:

We have to find the intervals in which function is increasing and decreasing.

Hint:

Put f ‘(x) = 0 to find the critical points and then use increasing and decreasing property.

Solution:

We have,

Differentiating w.r.t. x we get,

For f(x), we have to find critical points.

We must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xiii

Answer:Given:

Here given that

To find:

We have to find the intervals in which function is increasing and decreasing.

Hint:

Firstly, we will find critical points and then use increasing and decreasing property.

Soution:

We have,

Differentiating w.r.t. x, we get,

For critical points we must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xiv

Answer:Given:

Here given that

To find:

We have to find the increasing and decreasing intervals of f(x).

Hint:

Put f ‘(x) = 0 to find the critical points.

Solution:

We have,

On differentiating we get,

For f(x), we have to find critical points.

We must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xv

Answer:Given:

Here given that

To find:

We have to find the intervals in which f(x) is increasing or decreasing.

Hint:

Use product rule of differentiation

Solution:

We have,

Differentiating w.r.t. x, we get,

For f(x) we have to find critical points, for this we must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xvi

Answer:Given:

Here given that

To find:

We have to find the intervals in which f(x) is increasing or decreasing.

Hint:

Put f ‘(x) = 0 to find critical points and then use increasing and decreasing property.

Solution:

We have,

Differentiating w.r.t. x, we get,

For f(x) we have to find critical points, for this we must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xvii

Answer:Given:

Here given that

To find:

We have to find the intervals in which f(x) is increasing or decreasing.

Hint:

First we will find critical points and then use increasing and decreasing property.

Solution:

We have,

Differentiating w.r.t. x, we get,

For f(x) we have to find critical points,

We must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xviii

Answer:Given:

Here given that

To find:

We have to find the intervals in which f(x) is increasing or decreasing.

Hint:

Put f ‘(x) = 0 to find critical points and then use increasing and decreasing property.

Solution:

We have,

Differentiating w.r.t. x, we get,

For f(x) we have to find critical points

we must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xix

Answer:Given:

Here given that

To find:

We have to find the intervals in which f(x) is increasing and decreasing.

Hint:

We will find critical points.

Solution:

We have,

Differentiating w.r.t. x, we get,

For critical points we must have,

Taking cube root on both sides.

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xx

Answer:Given:

Here given that

To find:

We have to find the increasing or decreasing interval for f(x).

Hint:

First we will find critical points and then use increasing and decreasing property.

Solution:

We have,

Differentiating w.r.t. x, we get,

For critical points. we must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xxi

Answer:Given:

Here given that

To find:

We have to find the increasing or decreasing interval for f(x).

Hint:

Put f ‘(x) = 0 to find critical points and then use increasing and decreasing property.

Solution:

We have,

Differentiating w.r.t. x, we get,

For critical points. we must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xxii

Answer:Given:

Here given that

To find:

We have to find the increasing and decreasing intervals.

Hint:

We will find the critical points and then use increasing and decreasing property.

Solution:

We have,

Differentiating w.r.t. x we get,

For f(x), we have to find critical points.

We must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xxiii

Answer:Given:

Here given that

To find:

We have to find the increasing and decreasing intervals.

Hint:

First find the critical points by using f ‘(x) and then apply increasing and decreasing property.

Solution:

We have,

Differentiating w.r.t. x we get,

For critical points. We must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xxiv

Answer:Given:

Here given that

To find:

We have to find the intervals in which function is increasing or decreasing.

Hint:

Put f ‘(x) = 0 to find critical points of f(x) and then use increasing and decreasing property.

Solution:

We have,

Differentiating w.r.t. x, we get,

For f(x) we have to find critical points,

We must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xxv

Answer:Given:

Here given that

To find:

We have to find the increasing or decreasing interval for f(x).

Hint:

Put f ‘(x) = 0 to find critical points and then use increasing and decreasing property.

Solution:

We have,

Differentiating w.r.t. x, we get,

For f(x), we have to find critical points.

we must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xxvi

Answer:Given:

Here given that

To find:

We have to find the increasing and decreasing interval.

Hint:

Solution:

We have,

Differentiating w.r.t. x, we get,

Now,

For critical points, we must have,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xxvii

Answer:Given:

Here given that

To find:

We have to find the intervals in which function is increasing and decreasing.

Hint:

Solution:

We have,

Differentiating w.r.t. x, we get,

We have to find critical points, we must have,

The points x=0, 5, -3 divide the real line into four disjoint intervals,

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xxviii

Answer:

Given:

Here given that

To find:

We have to find the increasing and decreasing intervals of f(x).

Hint:

Solution:

We have,

Differentiating w.r.t. x we get,

For critical points. We must have,

Thus, f(x) is increasing on the interval (2,)

So,f(x) is decreasing on the interval (,2).

Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xxvix

Answer:Given:

Here given that

To find:

We have to find the intervals in which function is increasing and decreasing.

Hint:

Put f ‘(x) = 0 to find critical points and then use increasing and decreasing property.

Solution:

We have,

Differentiating w.r.t. x, we get,

At critical points,

The points x=-3, 2, 4 divide the real line into four disjoint intervals,

Increasing and Decreasing Functions exercise 16.2 question 2

Answer:Given:

To find:

We have to find the value of x, also we have to find co-ordinate of the point on the given curve where the normal is parallel to the line y = x+5.

Hint:

- First we will find critical point then find f’(x) for all the value of x.
- Find m
_{1}and m_{2}from the curve then find points

We have

For f(x) let us find critical point, we must have

Thus f(x) increases on 3, and f(x) is decreasing on interval x ∈ (-, 3)

Now,let us find co-ordinates of point on the equation of curve is

Slope of this curve is given by

Equation of line is y=x+5

Slope of this curve is

Since slope of curve is parallel to line.

Therefore, they follow the relation

Thus putting the value of x in equation of curve

We get

Hence the required co-ordinate is

Increasing and Decreasing Functions exercise 16.2 question 3

Answer:Given:

To find:

We have to interval in which f(x) is increasing or decreasing.

Hint:

f'(x)=0 to find critical point then use increasing or decreasing property.

Solution:

We have

On differentiating both sides we get

For f(x) let us find critical point, we must have

On dividing by cos x we get

Here the points divide the angle range from 0 to 2π.

Since we have x as angle.

Increasing and Decreasing Functions exercise 16.2 question 7

Answer:Given:

To show:

Hint:

Solution:

Given

On differentiating both sides w.r.t x we get

Taking different region from 0 to 2π.

Let

Therefor the above condition we find that

Hence,f(x) is neither increasing nor decreasing in (0, π)

Increasing and Decreasing Functions exercise 16.2 question 8

Answer:Given:

To show:

Hint:

Use increasing and decreasing property to find increasing and decreasing.

Solution:

Given

On differentiating both sides w.r.t x we get

Taking different region from 0 to

Hence proved.

Increasing and Decreasing Functions exercise 16.2 question 9

Answer:f(x) is increasing on interval

Given:

To prove:

We have to show that f(x) is increasing for all

Hint:

Show f’(x)>0 for f(x) to be increasing.

Solution:

Given

On differentiating both sides w.r.t x we get

Now, as given

Hence condition of f(x) to be increasing.

Thus f(x) is increasing on interval

Increasing and Decreasing Functions exercise 16.2 question 10

Answer:f(x) is increasing for all

Given:

To prove:

We have to show that f(x) is increasing for all

Hint:

Show f’(x)>0 for f(x) to be increasing.

Solution:

Given

On differentiating both sides w.r.t x we get

Now, as given

Hence condition for f(x) to be increasing.

Thus f(x) is increasing on interval

Increasing and Decreasing Functions exercise 16.2 question 11

Answer:Given:

To prove:

Hint:

Condition for f(x) to be decreasing that f’(x)<0.

Solution:

Given

On differentiating both sides w.r.t x we get

Now, as given

By applying negative sign, change in comparison sign.

Hence condition of f(x) to be decreasing.

Thus f(x) is decreasing on interval

.

Increasing and Decreasing Functions exercise 16.2 question 12

Answer:Given:

To prove:

Hint:

Condition for f(x) to be increasing that f’(x)>0.

Solution:

Given

On differentiating both sides w.r.t x we get

Hence condition of f(x) to be increasing.

Increasing and Decreasing Functions exercise 16.2 question 13

Answer:f(x) is decreasing on (0,) and increasing on (-,0) and neither increasing nor decreasing on (-,).

Given:

To show:

We have to show that f(x) is decreasing on (0,) and increasing on (-,0) and neither increasing nor decreasing on (-,).

Hint:

- For increasing function f'(x)>0 then f(x) is increasing.
- For f(x) to be decreasing f'(x)<0

Given

On differentiating both sides w.r.t x we get

Taking different region from - to .

By applying negative sign change comparison sign.

Thus f(x) is decreasing (0,)

By applying negative sign change comparison sign.

Thus f(x) is increasing (-,0).

Therefore from above condition we find that

?f(x) is decreasing in (0, ) and increasing in (-,0)

Hence, f(x) is neither increasing nor decreasing in (-, )

Increasing and Decreasing Functions exercise 16.2 question 14

Answer:Given:

To prove:

Hint:

Condition for increasing function f’(x)>0.

Solution:

Given

On differentiating both sides w.r.t x we get

Now, as given

i.e, 4

Hence condition of f(x) to be increasing.

Hence proved.

Increasing and Decreasing Functions exercise 16.2 question 15

Answer:Given:

To prove:

Hint:

Condition for decreasing function f’(x)<0.

Solution:

Given

On differentiating both sides w.r.t x we get

Now, as given

as here cosine values are smaller than sine values for same angle.

Hence condition for f(x) to be decreasing.

as here cosine values are smaller than sine values for same angle.

Hence condition for f(x) to be decreasing.

Increasing and Decreasing Functions exercise 16.2 question 16

Answer:Given:

To prove:

Hint:

Condition for f(x) to be decreasing is f’(x)<0.

Solution:

Given

On differentiating both sides w.r.t x we get

Now, as given

Hence condition for f(x) to be decreasing.

Increasing and Decreasing Functions exercise 16.2 question 17

Answer:Given:

To prove:

Hint:

- condition for f(x) to be increasing is f'(x)>0
- condition for f(x) to be decreasing f'(x)<0

Given

On differentiating both sides w.r.t x we get

Now, as given

Increasing and Decreasing Functions exercise 16.2 question 18

Answer:Given:

To prove:

Hint:

f’(x) > 0 is condition for increasing function of f(x).

Solution:

Given

On differentiating both sides w.r.t x we get

Hence condition of f(x) to be increasing is satisfied.

Thus f(x) is increasing on interval x>0

Increasing and Decreasing Functions exercise 16.2 question 19

Answer:f(x) is neither decreasing nor increasing on (0,1)

Given:

To prove:

We have to show that f(x) is neither decreasing nor increasing on (0,1).

Hint:

- condition for f(x) to be increasing is f'(x)>0
- condition for f(x) to be decreasing is f'(x)<0

Given

On differentiating both sides w.r.t x we get

Taking different region from (0,1)

Therefor from above condition we find that

Hence, f(x) is neither increasing nor decreasing in (0,1)

Increasing and Decreasing Functions exercise 16.2 question 20

Answer:Given:

To prove:

Hint:

f’(x) > 0 is condition for increasing function of f(x).

Solution:

Given

On differentiating both sides w.r.t x we get

Hence, condition for fx to be increasing

Increasing and Decreasing Functions exercise 16.2 question 21

Answer:f(x) is an increasing on R.

Given:

To prove:

We have to prove that f(x) is an increasing on R.

Hint:

Show f’(x) > 0 for increasing function.

Solution:

Given

On differentiating both sides w.r.t x we get

Thus f(x) is increasing function x R

Increasing and Decreasing Functions exercise 16.2 question 22

Answer:f(x) is an increasing on the interval [4, 6].

Given:

To prove:

We have to prove that f(x) is an increasing on the interval [4, 6].

Hint:

A function f(x) is said to be increasing on [a, b] if f’(x) > 0.

Solution:

Given

On differentiating both sides w.r.t x we get

Thus f(x) is an increasing function x [4, 6]

Increasing and Decreasing Functions exercise 16.2 question 23

Answer:Given:

To prove:

Hint:

If f ’(x) > 0 ∀ x (a,b) then f(x) is increasing on (a,b).

Solution:

Given

On differentiating both sides w.r.t x we get

Now, as given

Increasing and Decreasing Functions exercise 16.2 question 24

Answer:f(x) is decreasing function on R

Given:

To prove:

We have to show that f(x) is decreasing function on R.

Hint:

A function f(x) to be decreasing if f ’(x) > 0

Solution:

We have

On differentiating both sides w.r.t x we get

By applying negative sign change comparison sign

Hence f(x) is decreasing function for x R

Increasing and Decreasing Functions exercise 16.2 question 25

Answer:Given:

To prove:

Hint:

For f(x) to be increasing we must have f ’(x) > 0 and f ‘(x) < 0 for decreasing.

Solution:

We have

On differentiating both sides w.r.t x we get

Increasing and Decreasing Functions exercise 16.2 question 26

Answer:f(x) is increasing in (0,)

and f(x) is decreasing (-1,0)

Given:

To find:

We have to find the intervals in which f(x) is increasing and decreasing.

Hint:

First, we find critical point then use property of increasing and decreasing.

Solution:

We have,

Differentiating w.r.t. x we get,

For critical points. We must have,

Hence,f(x) is increasing in (0,), decreases in (-1,0).

Increasing and Decreasing Functions exercise 16.2 question 27

Answer:f(x) is increasing in (-,-1)

and f(x) is decreasing in (-1,)

Given:

To find:

We have to find the intervals in which f(x) is increasing and decreasing.

Hint:

First, we find critical point then find property of increasing and decreasing intervals.

Solution:

we have

Differentiating w.r.t. x we get,

For critical points.

Hence, f(x) is increasing in (-,-1), decreasing in (-1,)

Increasing and Decreasing Functions exercise 16.2 question 28

Answer:f(x) is increasing for all x.

Given:

To find:

We have to show that f(x) is increasing for all x.

Hint:

Condition to be function is increasing i.e, f ‘(x) > 0

Solution:

We have,

On differentiating both sides w.r.t. x we get,

Hence, f(x) is an increasing function for all x.

Increasing and Decreasing Functions exercise 16.2 question 29

Answer:f(x)=x-[x] is increasing in (0,1).

Given:

To find:

We have to prove that f(x)=x-[x] is increasing in (0,1).

Hint:

For increasing function f ‘(x) > 0

Solution:

We have,

We know that

On differentiating both sides w.r.t. x we get,

Hence,f(x) is an increasing function for all x (0,1).

Increasing and Decreasing Functions exercise 16.2 question 29 textbook solutio

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Increasing and Decreasing Functions exercise 16.2 question 29

f(x)=x-[x] is increasing in (0,1).

Given:

To find:

We have to prove that f(x)=x-[x] is increasing in (0,1).

Hint:

For increasing function f ‘(x) > 0

Solution:

We have,

We know that

On differentiating both sides w.r.t. x we get,

Hence,f(x) is an increasing function for all x (0,1).

Increasing and Decreasing Functions exercise 16.2 question 30 subquestion (i)

Answer:f(x) is an increasing on R

Given:

To prove:

We have to prove that f(x) is an increasing on R.

Hint:

If f’(x) > 0 then f(x) is increasing function.

Solution:

Given

On differentiating both sides w.r.t x we get

Thus f(x) is increasing function for all x R.

Increasing and Decreasing Functions exercise 16.2 question 30 subquestion (ii)

Answer:f(x) is an increasing on R.

Given:

To prove:

We have to prove that f(x) is an increasing on R.

Hint:

If f’(x) > 0 then f(x) is increasing function.

Solution:

Here we have

On differentiating both sides w.r.t x we get

Thus f(x) is increasing function on R.

Increasing and Decreasing Functions exercise 16.2 question 31

Answer:Given:

To prove:

Hint:

- for f(x) to be increasing we must have f'(x)>0
- for f(x) to be decreasing we must have f'(x)<0

Given

On differentiating both sides w.r.t x we get

By applying negative sign change comparison sign

By applying negative sign change comparison sign

I

Increasing and Decreasing Functions exercise 16.2 question 32

Answer:f(x) is strictly increasing on R.

Given:

To prove:

We have to prove that f(x) is strictly increasing on R.

Hint:

For f(x) to be increasing we must have f’(x) > 0.

Solution:

Here we have

On differentiating both sides w.r.t x we get

Hence f(x) is strictly increasing function on R.

Increasing and Decreasing Functions exercise 16.2 question 33 subquestion (i)

Answer:f(x) is strictly decreasing on R

Given:

To prove:

We have to prove that f(x) is strictly decreasing on R.

Hint:

For f(x) to be decreasing we must have f’(x) < 0.

Solution:

Given

On differentiating both sides w.r.t x we get

By applying negative sign change comparison sign.

Hence f(x) is strictly decreasing (0, ).

Increasing and Decreasing Functions exercise 16.2 question 33 subquestion (ii)

Answer:f(x) is strictly increasing in (,2)

Given:

To prove:

We have to prove that f(x) is strictly increasing in (,2)

Hint:

For f(x) to be increasing we must have f’(x) > 0.

Solution:

Given

On differentiating both sides w.r.t x we get

By applying negative sign change comparison sign.

Hence f(x) is strictly increasing on (,2)

Increasing and Decreasing Functions exercise 16.2 question 33 subquestion (iii)

Answer:f(x) is neither increasing nor decreasing in (0,2)

Given:

To prove:

We have to prove that f(x) is neither increasing nor decreasing in (0,2)

Hint:

For f(x) to be increasing we must have f’(x) > 0

and for f(x) to be decreasing we must have f’(x) < 0

Solution:

Given

On differentiating both sides w.r.t x we get

By applying negative sign change comparison sign.

Hence f(x) is decreasing function in (0, )

Again,

By applying negative sign change comparison sign.

Clearly, from above we get f(x) is neither increasing nor decreasing in (0,2)

Increasing and Decreasing Functions exercise 16.2 question 34

Answer:Given:

To prove:

Hint:

For f(x) to be increasing we must have f’(x) > 0.

Solution:

Given

On differentiating both sides w.r.t x we get

Increasing and Decreasing Functions exercise 16.2 question 35

Given:

To prove:

We have to find the value of a for which f(x) is an increasing function on R.

Hint:

Given f(x) is increasing function that means f’(x) > 0.

Solution:

Here we have

On differentiating both sides w.r.t x we get

Given f(x) is increasing function on R

Hence a ≤ 0 is required value of a.

Increasing and Decreasing Functions exercise 16.2 question 35

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Increasing and Decreasing Functions exercise 16.2 question 35

Answer:Given:

To prove:

We have to find the value of a for which f(x) is an increasing function on R.

Hint:

Given f(x) is increasing function that means f’(x) > 0.

Solution:

Here we have

On differentiating both sides w.r.t x we get

Given f(x) is increasing function on R

Hence a ≤ 0 is required value of a.

Increasing and Decreasing Functions exercise 16.2 question 36

Answer:Given:

To prove:

We have to find the value of b for which f(x) is decreasing function on R.

Hint:

We will apply f’(x) < 0 for decreasing then evaluate the value of b.

Solution:

We have

On differentiating both sides w.r.t x we get

Given f(x) is decreasing function on R

But the least value of cos x is 1

Hence b≥1 is required value of b.

Increasing and Decreasing Functions exercise 16.2 question 37

Answer:f(x) is an increasing function on R or all the value of a

Given:

To prove:

We have to show that f(x) is an increasing function on R or all the value of a.

Hint:

For f(x) to be increasing function we must have f’(x) > 0.

Solution:

Here we have

On differentiating both sides w.r.t x we get

Then we have,

Hence the function is increasing on Rf or all value of a.

Increasing and Decreasing Functions exercise 16.2 question 38

Answer:Given:

To prove:

Hint:

Using mean value theorem

Solution:

As f(0) = f(1) and f is differentiable

Hence by rolls theorem

Let us now apply mean value theorem for point 0 and x [0,1]

Hence

Now, as both x and c lies in [0,1]

Increasing and Decreasing Functions exercise 16.2 question 39 subquestion (i)

Answer:f(x)=x |x| is an increasing function for all real values.

Given:

f(x)=x |x|

To prove:

We have to find the interval in which f(x) is an increasing or decreasing.

Hint:

For f(x) to be increasing function we must have f’(x) > 0.

Solution:

Here we have

We know that

On differentiating f(x) w.r.t x we get

Hence f(x) is increasing function for all value of real values.

Increasing and Decreasing Functions exercise 16.2 question 39 subquestion (ii)

Answer:Given:

To prove:

We have to find the interval in which f(x) is an increasing or decreasing.

Hint:

- for f(x) to be increasing we must have f'(x)>0
- for f(x) to be decreasing we must have f'(x)<0

Here we have

We know that

On differentiating f(x) w.r.t x we get

Therefore the function fx is neither increasing nor decreasing in the interval (,2)

Increasing and Decreasing Functions exercise 16.2 question 39 subquestion (iii)

Answer:Given:

To find:

We have to find the intervals in which f(x) is increasing or decreasing.

Hint:

- for f(x) to be increasing we must have f'(x) > 0
- for f(x) to be decreasing we must have f'(x) < 0

We have,

Differentiating w.r.t. x we get,

For f(x) to be increasing, we must have,

This can only be possible when,

For f(x) to be decreasing we must have,

RD Sharma Class 12 Solutions Increasing and Decreasing Functions exercise 16.2 should be thoroughly followed by all students if they want to have impressive scores. Students should start their preparations beforehand and practice on all days to score high grades and surpass their peers.

The book contains information on various mathematical formulae, issues, and solutions which will give students a clear understanding of all chapters. Consequently, the students use the data provided in the book to test their own scores and track their performance.

Rd Sharma class 12 chapter 16 exercise 16.2 will always have the latest syllabus present in NCERT Books and prescribed by CBSE. The book will further help students to learn new methods of solving questions and expand their knowledge on the subject. RD Sharma Class 12th Exercise 16.2 solution will have answers on the following concepts: -

The solution of judicious logarithmic disparities

Stringently increasing functions

Stringently decreasing functions

Monotonic functions

Monotonically increasing function

Monotonically decreasing functions

Fundamental and adequate conditions for monotonicity

Discovering the stretches in which a job is increasing or decreasing

Demonstrating the monotonicity of a situation on a given stretch

Discovering the stretch where a cycle is increasing or decreasing

Benefits of using Rd Sharma class 12 chapter 16 exercise 16.2

There are certain benefits that are associated with using RD Sharma solutions for exam preparations. Some of the advantages include:-

One book covers all chapters and includes answers to all questions provided in NCERT maths books.

Simple solutions given to help understand the ideas better.

Zero Cost and no investment required to avail the pdf of the book.

A detailed solving of problems and formulae to teach students the basics of maths.

Great for last minute revisions and to test self-performance.

- Chapter 1 - Relations
- Chapter 2 - Functions
- Chapter 3 - Inverse Trigonometric Functions
- Chapter 4 - Algebra of Matrices
- Chapter 5 - Determinants
- Chapter 6 - Adjoint and Inverse of a Matrix
- Chapter 7 - Solution of Simultaneous Linear Equations
- Chapter 8 - Continuity
- Chapter 9 - Differentiability
- Chapter 10 - Differentiation
- Chapter 11 - Higher Order Derivatives
- Chapter 12 - Derivative as a Rate Measurer
- Chapter 13 - Differentials, Errors and Approximations
- Chapter 14 - Mean Value Theorems
- Chapter 15 - Tangents and Normals
- Chapter 16 - Increasing and Decreasing Functions
- Chapter 17 - Maxima and Minima
- Chapter 18 - Indefinite Integrals
- Chapter 19 - Definite Integrals
- Chapter 20 - Areas of Bounded Regions
- Chapter 21 - Differential Equations
- Chapter 22 - Algebra of Vectors
- Chapter 23 - Scalar Or Dot Product
- Chapter 24 - Vector or Cross Product
- Chapter 25 - Scalar Triple Product
- Chapter 26 - Direction Cosines and Direction Ratios
- Chapter 27 - Straight Line in Space
- Chapter 28 - The Plane
- Chapter 29 - Linear programming
- Chapter 30- Probability
- Chapter 31 - Mean and Variance of a Random Variable

1. How can I download Sharma class 12th exercise 16.2 pdf?

It is pretty easy to download the RD Sharma Class 12th Exercise 16.2 pdf online. Students need to navigate to the webpage of Career360 to find the free pdf.

2. Is RD Sharma useful to score high in the board exams?

There are numerous exam resources that students can use for preparations. However, preparing with RD Sharma Book Solutions will help them the most to understand everything and score better in class 12 exams.

3. Which is the best site to get class 12 rd Sharma chapter 16 exercise 16.2 solution for Class 12?

Students should use the Career360 website to get the Rd Sharma class 12 chapter 16 exercise 16.2 pdf as it's the one-stop destination for all RD Sharma solutions.

4. Are the RD Sharma Class 12 Maths Solutions Chapter 16.2 sufficient for CBSE students?

It's significantly recommended that class 12 RD Sharma chapter 16.2 exercise 16.2 solution students pick the RD Sharma Class 12 Solutions from Career360 as reference material for solutions.

5. List the number of questions in class 12 RD Sharma chapter 16.2?

There are 70 questions and answers in the RD Sharma Solutions Class 12 RD Sharma chapter 16.2 exercise 16.2.

Application Date:20 November,2023 - 19 December,2023

Application Date:20 November,2023 - 19 December,2023

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Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.

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The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary.

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A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.

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If you are intrigued by the programming world and are interested in developing communications networks then a career as database architect may be a good option for you. Data architect roles and responsibilities include building design models for data communication networks. Wide Area Networks (WANs), local area networks (LANs), and intranets are included in the database networks. It is expected that database architects will have in-depth knowledge of a company's business to develop a network to fulfil the requirements of the organisation. Stay tuned as we look at the larger picture and give you more information on what is db architecture, why you should pursue database architecture, what to expect from such a degree and what your job opportunities will be after graduation. Here, we will be discussing how to become a data architect. Students can visit NIT Trichy, IIT Kharagpur, JMI New Delhi.

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The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.

Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.

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Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.

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The role of geotechnical engineer starts with reviewing the projects needed to define the required material properties. The work responsibilities are followed by a site investigation of rock, soil, fault distribution and bedrock properties on and below an area of interest. The investigation is aimed to improve the ground engineering design and determine their engineering properties that include how they will interact with, on or in a proposed construction.

The role of geotechnical engineer in mining includes designing and determining the type of foundations, earthworks, and or pavement subgrades required for the intended man-made structures to be made. Geotechnical engineering jobs are involved in earthen and concrete dam construction projects, working under a range of normal and extreme loading conditions.

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How fascinating it is to represent the whole world on just a piece of paper or a sphere. With the help of maps, we are able to represent the real world on a much smaller scale. Individuals who opt for a career as a cartographer are those who make maps. But, cartography is not just limited to maps, it is about a mixture of art, science, and technology. As a cartographer, not only you will create maps but use various geodetic surveys and remote sensing systems to measure, analyse, and create different maps for political, cultural or educational purposes.

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A career as Bank Probationary Officer (PO) is seen as a promising career opportunity and a white-collar career. Each year aspirants take the Bank PO exam. This career provides plenty of career development and opportunities for a successful banking future. If you have more questions about a career as Bank Probationary Officer (PO), what is probationary officer or how to become a Bank Probationary Officer (PO) then you can read the article and clear all your doubts.

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Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.

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The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.

Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.

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A career as a Finance Executive requires one to be responsible for monitoring an organisation's income, investments and expenses to create and evaluate financial reports. His or her role involves performing audits, invoices, and budget preparations. He or she manages accounting activities, bank reconciliations, and payable and receivable accounts.

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An Investment Banking career involves the invention and generation of capital for other organizations, governments, and other entities. Individuals who opt for a career as Investment Bankers are the head of a team dedicated to raising capital by issuing bonds. Investment bankers are termed as the experts who have their fingers on the pulse of the current financial and investing climate. Students can pursue various Investment Banker courses, such as Banking and Insurance, and Economics to opt for an Investment Banking career path.

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Bank Branch Managers work in a specific section of banking related to the invention and generation of capital for other organisations, governments, and other entities. Bank Branch Managers work for the organisations and underwrite new debts and equity securities for all type of companies, aid in the sale of securities, as well as help to facilitate mergers and acquisitions, reorganisations, and broker trades for both institutions and private investors.

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Treasury analyst career path is often regarded as certified treasury specialist in some business situations, is a finance expert who specifically manages a company or organisation's long-term and short-term financial targets. Treasurer synonym could be a financial officer, which is one of the reputed positions in the corporate world. In a large company, the corporate treasury jobs hold power over the financial decision-making of the total investment and development strategy of the organisation.

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A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

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A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.

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A Conservation Architect is a professional responsible for conserving and restoring buildings or monuments having a historic value. He or she applies techniques to document and stabilise the object’s state without any further damage. A Conservation Architect restores the monuments and heritage buildings to bring them back to their original state.

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A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.

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A Team Leader is a professional responsible for guiding, monitoring and leading the entire group. He or she is responsible for motivating team members by providing a pleasant work environment to them and inspiring positive communication. A Team Leader contributes to the achievement of the organisation’s goals. He or she improves the confidence, product knowledge and communication skills of the team members and empowers them.

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A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.

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Individuals in the architecture career are the building designers who plan the whole construction keeping the safety and requirements of the people. Individuals in architect career in India provides professional services for new constructions, alterations, renovations and several other activities. Individuals in architectural careers in India visit site locations to visualize their projects and prepare scaled drawings to submit to a client or employer as a design. Individuals in architecture careers also estimate build costs, materials needed, and the projected time frame to complete a build.

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Having a landscape architecture career, you are involved in site analysis, site inventory, land planning, planting design, grading, stormwater management, suitable design, and construction specification. Frederick Law Olmsted, the designer of Central Park in New York introduced the title “landscape architect”. The Australian Institute of Landscape Architects (AILA) proclaims that "Landscape Architects research, plan, design and advise on the stewardship, conservation and sustainability of development of the environment and spaces, both within and beyond the built environment". Therefore, individuals who opt for a career as a landscape architect are those who are educated and experienced in landscape architecture. Students need to pursue various landscape architecture degrees, such as M.Des, M.Plan to become landscape architects. If you have more questions regarding a career as a landscape architect or how to become a landscape architect then you can read the article to get your doubts cleared.

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An expert in plumbing is aware of building regulations and safety standards and works to make sure these standards are upheld. Testing pipes for leakage using air pressure and other gauges, and also the ability to construct new pipe systems by cutting, fitting, measuring and threading pipes are some of the other more involved aspects of plumbing. Individuals in the plumber career path are self-employed or work for a small business employing less than ten people, though some might find working for larger entities or the government more desirable.

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Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.

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A veterinary doctor is a medical professional with a degree in veterinary science. The veterinary science qualification is the minimum requirement to become a veterinary doctor. There are numerous veterinary science courses offered by various institutes. He or she is employed at zoos to ensure they are provided with good health facilities and medical care to improve their life expectancy.

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A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.

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Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth.

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When it comes to an operation theatre, there are several tasks that are to be carried out before as well as after the operation or surgery has taken place. Such tasks are not possible without surgical tech and surgical tech tools. A single surgeon cannot do it all alone. It’s like for a footballer he needs his team’s support to score a goal the same goes for a surgeon. It is here, when a surgical technologist comes into the picture. It is the job of a surgical technologist to prepare the operation theatre with all the required equipment before the surgery. Not only that, once an operation is done it is the job of the surgical technologist to clean all the equipment. One has to fulfil the minimum requirements of surgical tech qualifications.

**Also Read:** Career as Nurse

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An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.

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Are you searching for a chemical pathologist job description? A chemical pathologist is a skilled professional in healthcare who utilises biochemical laboratory tests to diagnose disease by analysing the levels of various components or constituents in the patient’s body fluid.

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A Biochemical Engineer is a professional involved in the study of proteins, viruses, cells and other biological substances. He or she utilises his or her scientific knowledge to develop products, medicines or ways to improve quality and refine processes. A Biochemical Engineer studies chemical functions occurring in a living organism’s body. He or she utilises the observed knowledge to alter the composition of products and develop new processes. A Biochemical Engineer may develop biofuels or environmentally friendly methods to dispose of waste generated by industries.

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For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs.

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Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.

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Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages. Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.

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The career as a Talent Agent is filled with responsibilities. A Talent Agent is someone who is involved in the pre-production process of the film. It is a very busy job for a Talent Agent but as and when an individual gains experience and progresses in the career he or she can have people assisting him or her in work. Depending on one’s responsibilities, number of clients and experience he or she may also have to lead a team and work with juniors under him or her in a talent agency. In order to know more about the job of a talent agent continue reading the article.

If you want to know more about talent agent meaning, how to become a Talent Agent, or Talent Agent job description then continue reading this article.

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Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.

A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.

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A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.

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The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.

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Individuals who opt for a career as a talent director are professionals who work in the entertainment industry. He or she is responsible for finding out the right talent through auditions for films, theatre productions, or shows. A talented director possesses strong knowledge of computer software used in filmmaking, CGI and animation. A talent acquisition director keeps himself or herself updated on various technical aspects such as lighting, camera angles and shots.

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In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.

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Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.

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For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.

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In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion. Ever since internet cost got reduced the viewership for these types of content has increased on a large scale. Therefore, the career as vlogger has a lot to offer. If you want to know more about the career as vlogger, how to become a vlogger, so on and so forth then continue reading the article. Students can visit Jamia Millia Islamia, Asian College of Journalism, Indian Institute of Mass Communication to pursue journalism degrees.

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Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.

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Content writing is meant to speak directly with a particular audience, such as customers, potential customers, investors, employees, or other stakeholders. The main aim of professional content writers is to speak to their targeted audience and if it is not then it is not doing its job. There are numerous kinds of the content present on the website and each is different based on the service or the product it is used for.

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Individuals who opt for a career as a reporter may often be at work on national holidays and festivities. He or she pitches various story ideas and covers news stories in risky situations. Students can pursue a BMC (Bachelor of Mass Communication), B.M.M. (Bachelor of Mass Media), or MAJMC (MA in Journalism and Mass Communication) to become a reporter. While we sit at home reporters travel to locations to collect information that carries a news value.

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Linguistic meaning is related to language or Linguistics which is the study of languages. A career as a linguistic meaning, a profession that is based on the scientific study of language, and it's a very broad field with many specialities. Famous linguists work in academia, researching and teaching different areas of language, such as phonetics (sounds), syntax (word order) and semantics (meaning).

Other researchers focus on specialities like computational linguistics, which seeks to better match human and computer language capacities, or applied linguistics, which is concerned with improving language education. Still, others work as language experts for the government, advertising companies, dictionary publishers and various other private enterprises. Some might work from home as freelance linguists. Philologist, phonologist, and dialectician are some of Linguist synonym. Linguists can study French, German, Italian.

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Production Manager Job Description: A Production Manager is responsible for ensuring smooth running of manufacturing processes in an efficient manner. He or she plans and organises production schedules. The role of Production Manager involves estimation, negotiation on budget and timescales with the clients and managers.

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A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

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A career as Production Engineer is crucial in the manufacturing industry. He or she ensures the functionality of production equipment and machinery to improve productivity and minimize production costs in order to drive revenues and increase profitability.

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An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process.

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Individuals who opt for a career as product designers are responsible for designing the components and overall product concerning its shape, size, and material used in manufacturing. They are responsible for the aesthetic appearance of the product. A product designer uses his or her creative skills to give a product its final outlook and ensures the functionality of the design.

Students can opt for various product design degrees such as B.Des and M.Des to become product designers. Industrial product designer prepares 3D models of designs for approval and discusses them with clients and other colleagues. Individuals who opt for a career as a product designer estimate the total cost involved in designing.

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A career as R&D Personnel requires researching, planning, and implementing new programs and protocols into their organization and overseeing new products’ development. He or she uses his or her creative abilities to improve the existing products as per the requirements of the target market.

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A Commercial Manager negotiates, advises and secures information about pricing for commercial contracts. He or she is responsible for developing financial plans in order to maximise the business's profitability.

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ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks.

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Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack

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Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

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Individuals in the computer systems analyst career path study the hardware and applications that are part of an organization's computer systems, as well as how they are used. They collaborate closely with managers and end-users to identify system specifications and business priorities, as well as to assess the efficiency of computer systems and create techniques to boost IT efficiency. Individuals who opt for a career as a computer system analyst support the implementation, modification, and debugging of new systems after they have been installed.

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A Test Manager is a professional responsible for planning, coordinating and controlling test activities. He or she develops test processes and strategies to analyse and determine test methods and tools for test activities. The test manager jobs involve documenting tests that have been carried out, analysing and evaluating software quality to determine further recommended procedures.

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A career as Azure Developer comes with the responsibility of designing and developing cloud-based applications and maintaining software components. He or she possesses an in-depth knowledge of cloud computing and Azure app service.

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A Deep Learning Engineer is an IT professional who is responsible for developing and managing data pipelines. He or she is knowledgeable about analyzing and storing data collected from various sources. A Career as a Deep Learning Engineer needs to help the data scientists and analysts to create effective data sets.

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