Mesaurement and computer instrumentation. Digital to Analouge converters. 3. Figure 4.44 shows the graphs of four functions: Which graph corresponds to each function? 4. Repeat Exercise 2 for the...

Sol: (3) (a)   2 ,1
x
f x
x


(b)  
3
4
,
1
x
g x
x


(c)  
3
6
,
1
x x
h x
x



(d)  
3
4
,
1
x
k x
x


Sol: (2)
The given function
   2 on ,0f x x  
First we evaluate the local extreme. Take the first derivative of the given function,
 ' 1f x 
But there is no place where  ' 0f x  , so we don’t have any local extreme to be
concerned about.
That means we only evaluate at the end points.

 
 0 2
f
f
  


So the absolute maximum is 2.
And the absolute minimum is .
Hence the function has only absolute extreme values.
Sol: (20)
The given function
   
2
2 4 ,f x x 
First we evaluate the local extreme. Take the first derivative of the given function,
    2' 2 2 4f x x x 
To find the local extreme values,  ' 0f x 
  
  
2
2
2 2 4 0
2 4 0
0,2, 2
x x
x x
x
 
 
 

Now obtain the function values,
At 0,x 
   
   
 
2
2
2
4
0 0 4
0 16
f x x
f
f
 
 


At 2,x  

   
 
2
2 4 4
2 0
f
f
  


At 2,x 

   
 
2
2 4 4
2 0
f
f
 

Sol: (34)
The given function
 
22 ,xf x x e
First we evaluate the local extreme. Take the first derivative of the given function,

     
   
2 2
2
2
2
' 2 2
' 2 1
x x
x
f x x e x e x
f x xe x
 

  
 

To find the local extreme values,  ' 0f x 
 
 
2
2
2
2
2 1 0
1 0
1,0,1
x
x
xe x
xe x
x


 
 
 

Now obtain the extreme values,
At 1,x  
 
     
 
2
2
2
2 1
1
1 1
1 0.368
xf x x e
f e
f e

 


  
  

At 0,x 

 
 
22
0 0
xf x x e
f



At 1,x 
 
     
 
2
2
2
2 1
1
1 1
1 0.368
xf x x e
f e
f e





 

So these are all the local extreme values of the given function.
Sol: (6)
The given function is
  3 510 3f x x x 
On taking a first derivative of  f x ,
  2 4' 30 15f x x x 
On taking a second derivative of  f x ,
...