Concept explainers
a.
To find the intervals on which the function is increasing by using analytical method.
a.
Answer to Problem 11RE
The function
Explanation of Solution
Given:
The function is
Calculation:
Since,
At
The function is increasing when
Now , put
Now , put
Therefore, the function
Below is the graph of the function
From graph it is clear that the function
b.
To find the intervals on which the function is decreasing by using analytical method.
b.
Answer to Problem 11RE
The function
Explanation of Solution
Given:
The function is
Calculation:
Since,
At
The function is increasing when
Now , put
Now , put
Therefore, the function
Below is the graph of the function
From graph it is clear that the function
c.
To find the intervals on which the function is concave up by using analytical method.
c.
Answer to Problem 11RE
The Function
Explanation of Solution
Given:
The function is
Calculation:
The graph of a twice differentiable function
Concave up on any interval where
Since,
Second derivative:
Now, put
At
Now in interval
Now in interval
Therefore, the Function
Below is the graph of the function
From graph it is clear that ,the Function
d.
To find the intervals on which the function is concave down by using analytical method.
d.
Answer to Problem 11RE
Explanation of Solution
Given:
The function is
Calculation:
The graph of a twice differentiable function
Concave up on any interval where
Since,
Second derivative:
Now, put
At
Now in interval
Now in interval
Therefore, the Function
Below is the graph of the function
From graph it is clear that , the Function
e.
To find any local extreme values.
e.
Answer to Problem 11RE
Local extreme values exist at point
Explanation of Solution
Given:
The function is
Calculation:
Since, in the interval
Maximum values are
Therefore, local extreme values exist at point
f.
To find inflections points.
f.
Answer to Problem 11RE
No inflection point exist because function does not changes its concavity.
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,
Chapter 5 Solutions
Calculus: Graphical, Numerical, Algebraic
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