Concept explainers
a.
To find the intervals on which the function is increasing by using analytical method.
a.
Answer to Problem 9RE
There are no values of
Explanation of Solution
Given:
The function is
Calculation:
The function is increasing when
Since
Therefore,
So,
It means
Below is graph of function
From above graph it is clear that function
Hence , there are no values of
b.
To find the intervals on which the function is decreasing by using analytical method.
b.
Answer to Problem 9RE
Explanation of Solution
Given:
The function is
Calculation:
The function is decreasing when
Since
Therefore,
So,
It means
Below is graph of function
From above graph it is clear that function
c.
To find the intervals on which the function is concave up by using analytical method.
c.
Answer to Problem 9RE
The graph of function is concave up when
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 :
Now, put
Domain of
Therefore, there are two intervals that is
check the value of
Now for
Hence
Below is graph of function
From above graph it is clear that function
d.
To find the intervals on which the function is concave down by using analytical method.
d.
Answer to Problem 9RE
The graph of function is concave down when
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 :
Now, put
Domain of
Therefore, there are two intervals that is
On checking value of
Now for
Hence
Below is graph of function
From above graph it is clear that function
e.
To find any local extreme values.
e.
Answer to Problem 9RE
Explanation of Solution
Given:
The function is
Calculation:
Graph of
From graph it is clear that
f.
To find inflections points.
f.
Answer to Problem 9RE
The inflection point of
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.
Graph of
From graph it is clear that concavity of
The inflection point of
Chapter 5 Solutions
Calculus: Graphical, Numerical, Algebraic
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