------------------------------------- Increasing & Decreasing Functions / Concavity--------------------------------------1. (I 11/12a S) Given that (2,0) is a stationary point of the curve y  x  6 x  ax  b . Find
(i) the value of 𝑎 ;
4
2
[8]
 , 2  decreasing 


 2,   increasing 
 , 1  1,   concave upwards 
(iii) the interval(s) where the function is concave down or concave up. 

 1,1 concave downwards

(ii) the interval(s) where the function is increasing or decreasing;
2. (I 12/7c S) Given that the function f  x   4 x3  24 x 2  36 x  7 .
(i) Find f '( x) and f ''( x) ;
[ 12x2  48x  36;24x  48 ]
(1, 23);(3,7)
(ii) Find the coordinates of the stationary points of the function.
(iii) Hence, determine whether each stationary points is a local maximum point or a local minimum point.
1,23 max 


 3,7  min 
3. (I 13/12b) Find the intervals where the function f ( x) 
1 3
x  2 x2  3x is increasing or decreasing.
3
 ,1   3,   increasing 


1,3 decreasing

4.
(I 14/11b) Given the function y  f ( x)  2 x  9 x  12 x  3 . Find
(i) the coordinates of the stationary point(s);
3
2
(1,2),(2,1)
(ii) the respective intervals where the function is increasing and where it is decreasing.
 ,1   2,   increa sin g 


(1,2)decrea sin g

5.
(I15/11b) Given curve y  f ( x) passes through the point (0, 1) and
dy
4
 5  2 x  1  80 .
dx
1
1
5
 2 x  1  80 x  ]
2
2
(ii) Find the coordinates of the local extreme points of the function y  f ( x) , and determine whether each
(i) Find 𝑓(𝑥).
of them is a local maximum or a local minimum point.
[ f ( x) 
 3 209 

  2 , 2  max 



 1 47 

 ,   min 
2 
 2

6. (I 16/12a) Find the global maximum value and global minimum value of the f ( x) 
x
on the interval
x 4
2
1
1

 4 max;  5 min 
 1,3 .
2 3 x
7. (I 18/5) Given the function f ( x)  x e
 2
0, 3 
, find the interval(s) on which 𝑓 is increasing.
x
has a local extreme point at 𝑥 = 3, find the values of 𝑎 and
x  a2
determine whether the extreme point is a local maximum point or a local minimum point.
[ a  3;max ]
8. (Is 19/5) If the function f ( x) 
2
9. (06/10 O) If function y  2 x  3(a  1) x  6ax  6 x  3 is an increasing function in the interval
3
2
1,5
 ,   , find the range of values of a .
10. (15/11a S) Given a function f ( x)  x x , x  0 ,
Find (i) f '( x) ;
(ii) the intervals at which f ( x) in increasing or decreasing.
1
[ xx 
11.
(14/MCQ17) Given that the inflection point of curve y 
B
A 1
12.
1  ln x
,  0,e  is increasing,  e,   is decreasing]
x2
3
2
2
C e3
ln x
is (a, b) , find a .
x
E
3
D e
E e2
  
.
,
 2 3 
(14/10b) Given f ( x)  3cos x  cos3x where the domain is  
(i) Find the stationary points of 𝑓(𝑥).
 
 

(0, 2),   , 2 2  ,  , 2 2 
 4
 4

(ii) Find the global maximum and global minimum values of 𝑓(𝑥).
[ 2 2,0 ]
13.
(15/MCQ 16) Find the coordinates of the inflection point(s) of the curve y  5x 4  x5 .
14.
(15/11a) Given function f ( x)  x x , 𝑥 > 0.
(i) Find f '( x) .
1  ln x
x2
 0, e increa sin g 


 e,   decrea sin g 
x
(ii) Find the interval at which 𝑓(𝑥) is increasing or decreasing.
15.
 3,162 
x
(16/9) Given f ( x)  x ln x , 𝑥 > 0.
2
(i) Find lim f ( x) .
x0
[0]
(ii) Find f '  x  and f ''  x  .
[ 3  2ln x ]
 1 

 0,
 decrea sin g 
e


 1


,   increa sin g 

 e 

(iii) Determine the interval at which 𝑓(𝑥) is increasing and decreasing.
1

 min :  2e 
Hence, find the local extreme value of 𝑓(𝑥), and determine its nature.
(iv) Determine the concavity of 𝑓 and its inflection point(s).
2 x






3
 0, e 2 concave downwards 


 32

 e ,  concave upward



 e 3 2 ,  3 e3  inflection point 



2 
16.
(17/MCQ19) Given that the function y  x e
A 0
B 1
17.
(17/9a) Given that the function f  x   2 x  ax  bx  5 is decreasing on the interval (2,5) and is
3
has a local maximum at x  a , find 𝑎.
C 2
D 3
2
increasing on the intervals  , 2  and  5,  . Find the values of a and b .
18.
C
 21;60
(19/10a) Given the function 𝑓(𝑥) = 2𝑥 + cos 𝑥.
(i) Show that 𝑓(𝑥) is an increasing function.
(ii) Find all the points of inflection on the curve 𝑦 = 2𝑥 + cos 𝑥.




 n  ,  2n  1   , n  Z 
2


