Concept explainers
a.
To find the intervals on which the function is increasing by using analytical method.
a.
Answer to Problem 12RE
The function
Explanation of Solution
Given:
The function is
Calculation:
The function is increasing when
Below is the graph of
From graph it can be observed that there are total eight critical points that is
Now ,put
Therefore
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Therefore
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Therefore
Now ,put
Therefore
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Therefore
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Therefore
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Therefore
Now ,put
Therefore
hence, the function
b.
To find the intervals on which the function is decreasing by using analytical method.
b.
Answer to Problem 12RE
The function
Explanation of Solution
Given:
The function is
Calculation:
The function is decreasing when
Below is the graph of
From graph it can be observed that there are total eight critical points that is
Now ,put
Therefore
Now ,put
Therefore
Now ,put
Therefore
Now ,put
Therefore
Now ,put
Therefore
Now ,put
Therefore
Now ,put
Therefore
Now ,put
Therefore
hence, the function
c.
To find the intervals on which the function is concave up by using analytical method.
c.
Answer to Problem 12RE
The function
Explanation of Solution
Given:
The function is
Calculation:
The graph of a twice differentiable function
Concave up on any interval where
Since,
First derivative :
Second derivative :
below is the graph of
From graph it is clear that , there are total eight critical points that are
Now, put
Therefore,
Now, put
Therefore,
Now, put
Therefore,
Now, put
Therefore,
Now, put
Therefore,
Now, put
Therefore,
Now, put
Therefore,
Now, put
Therefore,
Now, put
Therefore,
Hence, the function
d.
To find the intervals on which the function is concave down by using analytical method.
d.
Answer to Problem 12RE
The function
Explanation of Solution
Given:
The function is
Calculation:
The graph of a twice differentiable function
Concave up on any interval where
Since,
First derivative :
Second derivative :
below is the graph of
From graph it is clear that , there are total eight critical points that are
Now, put
Therefore,
Now, put
Therefore,
Now, put
Therefore,
Now, put
Therefore,
Now, put
Therefore,
Now, put
Therefore,
Now, put
Therefore,
Now, put
Therefore,
Now, put
Therefore,
Hence, the function
e.
To find any local extreme values.
e.
Answer to Problem 12RE
Explanation of Solution
Given:
The function is
Calculation:
Below is the graph of
From graph it is clear that
f.
To find inflections points.
f.
Answer to Problem 12RE
The inflection points are
Explanation of Solution
Given:
The function is
Calculation:
Inflection point of any function is a point where the graph of function has a tangent line and where the concavity changes.
Since, the intervals in which function is concave up are
The intervals in which function is concave down are
Therefore, the inflection points are
Chapter 5 Solutions
Calculus: Graphical, Numerical, Algebraic: Solutions Manual
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