ZERO LEVEL
Increasing and Decreasing Function – 01
1.
Show that the function f ( x )  e x is strictly increasing on
R.
2.
Show that the function f ( x )  5 x  2 is a strictly
increasing on R.
3.
13.
3
Show that the function f ( x )  x  6 x  12 x  18 is an
increasing function on R.
5.
Show that f ( x )  e1 / x is a strictly decreasing function
Prove that
15.
Find the intervals on which the function
f ( x )  2 x 3  15 x 2  36 x  6 is ,
(i) increasing,
16.
17.
1
is increasing for all x  R , where
x
(ii) decreasing .
Find the intervals on which the function
f ( x )  log(1  x ) 
(b) strictly decreasing on ( , 0 ) 
8.
4x 2  1
, x  0 is ,
x
(i) increasing,
Show that the function f ( x )  x is ,
(a) strictly increasing on (0,  ) ,
(ii) decreasing .
Find the intervals on which the function
f (x) 
for all x  0 
Show that f ( x )  x 
x
 log(1  x )  x for x  0 
1 x
14.
2
4.
7.
x
is ,
(1  x )
(i) increasing,
18.
x  0
f ( x )  2 x 3  3 x 2  36 x  7 is ,
Show that the function f ( x )  sin x is ,
(i) strictly increasing,
 
(a) strictly increasing on  0, 2 


19.
 
(b) strictly decreasing on  2 ,  


20.
Find the intervals on which the function f ( x ) 
Show that f ( x )  ( x  1) e x  1 is a strictly increasing
 
Show that f ( x )  log sin x is strictly increasing on  0, 2 


21.
(ii) decreasing .
(ii) decreasing .
Find the intervals on which the function
(i) strictly increasing,
22.
23.
(ii) strictly decreasing.
Find the intervals on which the function f ( x )  ( x  2) e  x
is,
(i) increasing,
(ii) decreasing
Show that the function f ( x )  ( x  x  1) is neither
Find the intervals on which the function
f ( x )  (sin x  cos), 0  x  2 ,
increasing nor decreasing on (0,1) 
(i) increasing,
2
Show that the function f ( x )  cos 2 x is strictly decreasing
 
on  0, 2  


24.
(ii) decreasing .
 
Separate 0, 2  into subintervals in which


f ( x )  sin 3 x is ,
(i) increasing,
09/II/13/ZLA-35/CAL-16/A/200
is,
f ( x )  2 x 3  9 x 2  12 x  1 is ,
 
and strictly decreasing on  2 ,   


12.
x
x2  1
Find the intervals on which the function
(i) increasing,
function for all x  0 
11.
(ii) strictly decreasing.
f ( x )  ( x  1)3 ( x  3 )3 is ,
(c) neither increasing nor decreasing on (0, ) 
10.
(ii) decreasing .
Find the intervals on which the function
(i) increasing,
9.
2x
is
x2
increasing for all x > – 1.
If a is a real number such that 0  a  1 show that the
function f ( x )  a x is strictly decreasing on R.
6.
Prove that the function f ( x )  log(1  x ) 
18, Zone-II, M.P. Nagar, Bhopal  : 0755-2768319
(ii) decreasing .
Assignment-035 |1
25.
 
Separate the interval 0,  into subintervals in which
 2
42.
1

function f ( x )   x   is strictly increasing on I.
x

f ( x )  (sin 4 x  cos 4 x ) is ,
(i) increasing,
26.
(ii) decreasing .
27.
(ii) decreasing .
Show that f ( x ) 
44.
1 

Show that f ( x )   x 3 
 is decreasing on (–1, 1).
x3 

45.
Show that the function f ( x )  ( x  sin x ) is increasing
Find the intervals on which the function
f ( x )  x 4  4 x 3  4 x 2  15 is increasing or
decreasing .
28.
Show that f ( x )  3 x  5 is a strictly increasing on R.
29.
Show that f ( x )  e  x is strictly decreasing on R.
30.
If a is a real number greater than 1, show that
32.
33.
Show that the function f ( x )  5 x  2 is a strictly
increasing function on R.
Show that the function f ( x )  2 x  7 is a strictly
decreasing function on R.
35.
Prove that function f ( x )  e 2 x is strictly increasing
on R.
47.
Show that f ( x ) 
48.
except at x  1
Find the intervals on which the function
Prove that f ( x )  3 x is strictly increasing on R.
37.
Prove that the function f ( x )  log a x is strictly increasing
on (0,  ) when a  1 and strictly decreasing on
(i) strictly increasing,
49.
Prove that the function f ( x )  log e x is strictly
increasing (0, ) 
39.
(i) strictly increasing,
50.
(i) strictly increasing on (0,  ) ,
(ii) strictly decreasing on (, 0 ) ,
(iii) neither strictly increasing nor strictly decreasing
on R.
(i) strictly increasing,
f ( x )  x 3  2 x 2  1 is,
(i) increasing,
(ii) decreasing .
Find the intervals on which the following functions, are :
(i) increasing, (ii) decreasing .
52.

x 3 
f (x)   x4 


3 

53.
f ( x )  x 3  3 x 2  105 x  25 
54.
f ( x )  5  36 x  3 x 2  2 x 3 
55.
f ( x )  2 x 3  9 x 2  12 x  15 
56.
f ( x )  x 3  6 x 2  9 x  10 
57.
f ( x )  6  12 x  3 x 2  2 x 3 
Show that f ( x )  x 3  15 x 2  75 x  50 is increasing on R.
58.
f ( x )  2 x 3  24 x  5 
41.
Show that f ( x )
3
 5 is decreasing for all x  R, where
x
59.
f ( x )  ( x  1)( x  2)2 
60.
f ( x )  x 4  2x 2 
2| Assignment-035
(ii) strictly decreasing.
Find the intervals on which the function
40.
x  0
(ii) strictly decreasing.
Find the intervals on which the function
f ( x )  6  9 x  x 2 is ,
2
Prove that the function f ( x )  x is :
(ii) strictly decreasing.
Find the intervals on which the function
f ( x )  2 x 2  3 x is ,
(0,  ) when 0  a  1.
38.
( x  2)
is increasing for all x  R ,
( x  1)
f ( x )  10  6 x  2 x 2 is ,
51.
36.


x

is increasing on  0,

2
sin x
Show that f ( x ) 
and a  0, is a strictly increasing function on R.
Show that the function f ( x )  e x is strictly increasing
on R.
is increasing for all x  0 
46.
Prove that f ( x )  ax  b , where a and b are constant
34.
(1  x 2 )
for all x  R.
f ( x )  a x is strictly increasing on R.
31.
1
43.
Find the values of x for which f ( x )  x x , x  0 is,
(i) increasing,
Let I be an interval disjoint from (–1, 1). Prove that the
18, Zone-II, M.P. Nagar, Bhopal  : 0755-2768319
Answers (Increasing and Decreasing Function)
48.
3 


(i)   ,

2 
 3

, 
(ii) 
 2

49.
3

(i)  , 
4

3

(ii)   , 

4
50.
9 


(i)   ,

2 
9

(ii)  , 
2

51.
4

(i)   ,   [0,  )

3
4 
(ii)  , 0
3 
52.
1

(i)  , 

4

1
(ii)    , 
4
53.
(i) ( ,  7]  [5,  )
(ii) [7, 5]
54.
(i) [–2, 3]
(ii) ( ,  2 ]  [3,  )
55.
(i) ( ,  2 ]  [ 1,  )
(ii) [–2, –1]
56.
(i) ( , 1]  [3,  )
(ii) [–1, 3]
57.
(i) [–1, 2]
(ii) ( ,  1]  [2,  )
58.
(i) (  ,  2 ]  [1,  )
(ii) [–2, 2]
59.
4

(i)   ,   [ 2,  )

3
4 
(ii)  , 2
3 
60.
Increasing on ( 1, 0 )  (1,  ) and decreasing on
( ,  1)  (0, 1) .
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Assignment-035 |3