1. [22 marks] Using the method outlined in Section 5.4 of the textbook, and showing the working, sketch the graph of each of the following functions. Note: you do not need to consider f ''(x),...

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Answered Same DayDec 22, 2021

Answer To: 1. [22 marks] Using the method outlined in Section 5.4 of the textbook, and showing the working,...

David answered on Dec 22 2021
129 Votes
Sol: 1(a)  
2
21( ) 4
4
f x x 


Domain: All real numbers.


intercept: 2x x  
intercept: 0, 1y x
y  
 
2
2
vertical asymptotes:
1
4
4
y x 

Symmetry:
    
    
   
2
2
2
2
1
4
4
1
4
4
f x x
f x x
f x f x
   
  
 

On taking a differentiation,
    
   
2
2
2
' 4 2
4
' 4
f x x x
f x x x
 
 

Again taking a differentiation

     
 
2
2
'' 4 2
'' 3 4
f x x x x
f x x
  
 
To find a critical point,
 ' 0f x 
    
 
2
2
2
' 4 2
4
4 0
2,0,2
f x x x
x x
x
 
 
 

See the critical point in below graph,
Sol: 1(b)
2
8
( )
5
f x
x x



 Domain: x R: 5 0 or 0x x    


intercept: 2x x  
intercept: 0, 1y x y  

2
2
vertical asymptotes:
8
as 5
5
8
as 0
5
x
x x
x
x x
 

 

Symmetry:
 
 
 
 
   
   
2
2
8
5
8
5
f x
x x
f x
x x
f x f x
or
f x f x
 
  
 
 
 
  

On taking a differentiation,
 
 
 
2
3 23
8
( )
5
20 4
'
5
f x
x x
x
f x
x x


 



Again taking a differentiation
 
 
 
2
5 24
10 7 56 120
''
5
x x
f x
x x
...
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