# Â  Â  Â  Â  Â  Â Exercise 6.2

class 12 maths ex 6.2 ncert solution

Question 1: Show that the function given by is strictly increasing on .

Solution:

Therefore, the function is strictly increasing on .

Question 2: Show that the function given by is strictly increasing on .

Solution:

, For all

Therefore, the function is strictly increasing on .

Question 3: Show that the function given by is

(a) strictly increasing in

(b) strictly decreasing in

(c) neither increasing nor decreasing in .

Solution :

(a) For all

Therefore, the function is strictly increasing on .

(b) , For all

Therefore, the function is strictly decreasing on .

(c) For all

and , For all

Therefore, the function is neither increasing nor decreasing on .

Question 4: Find the intervals in which the function given by is

(a) strictly increasing

(b) strictly decreasing

Solution :

If

(a) , For all ,

therefore, the function is strictly increasing on .

(b) , For all ,

therefore, the function is strictly decreasing on .

Question 5: Find the intervals in which the function given by is

(a) strictly increasing

(b) strictly decreasing

Solution:

Now

and divides the real number line into three intervals and .

(a) Function is increasing on and .

(b) Function is decreasing on .

Question 6: Find the intervals in which the following functions are strictly increasing or decreasing:

(a)

(b)

(c)

(d)

(e)

Solution: (a)

Now

, For all ,

therefore, the function is strictly decreasing on .

, For all ,

therefore, the function is strictly increasing on .

(b)

Now

, For all ,

therefore, the function is strictly increasing on .

, For all ,

therefore, the function is strictly decreasing on .

(c)

If

and divides the number line into three intervals and .

Function is increasing on and decreasing on .

(d)

Now

, For all ,

therefore, the function is strictly increasing on .

, For all ,

therefore, the function is strictly decreasing on .

(e)

Now

and divides the number lines into four intervals and .

Function is decreasing on and increasing on .

Question 7: Show that , is an increasing function of throughout its domain.

Solution :

and , as these are perfect square and

as .

Therefore, , if .

Hence, the function is increasing throughout its domain.

Question 8: Find the values of for which is an increasing function.

Solution:

Now

and divides the number line into four intervals and .

Function is increasing on .

Question 9: Prove that , is an increasing function of in .

Solution :

Therefore, , if .

Hence, is an increasing function of in .

Question 10: Prove that the logarithmic function is strictly increasing on .

Solution:

Therefore, the function is strictly increasing on .

Question 11: Prove that the function given by is neither strictly increasing nor strictly decreasing on .

Solution:

If

divides the interval into two intervals and .

, For all ,

therefore, the function is strictly decreasing on .

, For all , therefore, the function is strictly increasing on .

Therefore, the function is neither increasing nor decreasing on .

Question 12: Which of the following functions are strictly decreasing on ?

(A)

(B)

(C)

(D)

Solution: (A)

, For all ,

therefore, the function is strictly decreasing on .

(B)

If , then

if .

Therefore, if .

, For all ,

therefore, the function is strictly decreasing on .

(C)

If , then ,

if .

if or

, For all ,

therefore, the function is strictly decreasing on .

if ,

therefore, if or

, For all ,

therefore, the function is strictly increasing on .

Hence, the function is neither increasing nor decreasing on .

(D)

, For all ,

therefore, the function is strictly increasing on .

Therefore, and are decreasing on .

Hence, the option (A) and (B) are correct.

Question 13: On which of the following intervals is the function given by strictly decreasing?

(A)

(B)

(C)

(D) None of these

Solution:

(A) and , For all ,

therefore, the function is strictly increasing on .

(B) and , For all ,

, for all , .

Therefore, ,

therefore, the function is strictly increasing on .

(C) and , For all ,

therefore, the function is strictly increasing on .

Hence, is function is not decreasing in the given intervals. Hence, the option (D) is correct.

Question 14: Find the least value of such that the function given by is strictly increasing on .

Solution :

But is increasing on

Here, when or ,

Therefore, for the least value of , when the function is increasing on , we have

Question 15: Let I be any interval disjoint from . Prove that the function given by is strictly increasing on I.

Solution:

If

and divides the real value into intervals and .

Only and are disjoint from .

Function is increasing on .

Hence, the function is strictly increasing on interval I.

Question 16: Prove that the function given by is strictly increasing on and strictly decreasing on .

Solution:

, for all ,

therefore, the function is strictly increasing on .

, for all ,

therefore, the function is strictly decreasing on .

Question 17: Prove that the function given by is strictly decreasing on and strictly increasing on .

Solution:

, for all ,

therefore, the function is strictly decreasing on .

, for all ,

therefore, the function is strictly increasing on .

Question 18: Prove that the function given by is increasing in .

Solution :

, as it is perfect square.

Therefore, the function is increasing in .

Question 19: The interval in which is increasing is

(A)

(B)

(C)

(D)

Solution:Â

Now

and divides the real values into three intervals and .

Function is strictly increasing on .

Therefore, the option (D) is correct.