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| PAGE: 403 | SET: Exercises | PROBLEM: 78 |
(a)
Normal density function is 
.
Where 
is mean and positive constant 
is called standard deviation.
Let the function in special case by removing the factor 
and considering 
.
Therefore, the function is 
.
Horizontal asymptote :
Therefore 
is the horizontal asymptote of the function.
Maximum value:
Consider 
.
Differentiate on each side with respect to 
.
Find the critical points by equating the derivatve to zero.
The function has the maximum value at 
, Since 
.
Maximum value is 
.
Inflection points:
.
Differentiate on each side with respect to .
Find the inflection points by equating 
to zero.
Substitute 
in the function.
Inflection point is 
.
(b)
The curve equation is .
Property : 
, stretch the graph of the function 
horizontally by a factor of 
.
stretch the graph of the function 
horizontally by a factor of , for 
.
(c)
Graph :
Graph the curve for 
.
Observe the graph :
stretch horizontally as 
increases.
is the horizontal asymptote of the function.
Maximum value is .
Inflection point is 
.
stretch the graph of the function 
horizontally by a factor of , for 
.
stretch horizontally as increases.
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