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| PAGE: 214 | SET: Exercises | PROBLEM: 60 |
(a)
I/D test :
If 
is increasing on the interval, then 
.
If is decreasing on the interval, then 
.
At extrima values, the function 
.
Observe the graph of :
Local maximum occurs at 
.
Local minimum occurs at 
.
Therefore, the function 
for 
.
The function increases over the intervals 
and 
.
Therefore, 
for 
and 
.
The function decreases in the interval 
.
Therefore, 
for 
.
(b)
Observe the graph of :
The function is a third degree polynomial function with positive leading coefficient.
The derivative of a third degree function is a second degree function means 
is a quadratic function.
The derivative of a second degree function is a first degree function means 
is a linear function.
Therefore, 
for 
.
for 
.
for 
.
(c)
Since the function 
is a quadratic function, it is increasing on for 
.
Therefore, the function 
increases on 
.
(d)
Since the function is a quadratic function, the function is negative minimum for .
The rate of change of at is less than the rate of change of for all other values of 
.
The function is decreasing at the greatest rate at 
.
(a)
for .
for and .
for .
(b)
for .
for .
for 
.
(c)
The function 
increases on 
.
(d)
The function is minimum for .
The function is decreasing at the greatest rate at .
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