Concept explainers
a.
Tofindthe interval on which the function
a.
Answer to Problem 5RE
The function is then increasing for
Explanation of Solution
Given information:
The given function is
Formula:
Chain rule:
Consider the function
Using product and chain rule:
Here the second derivative is always negative, so the function is concave down for all values of
Here, the function is positive until one point which is at zero and then negative.
Therefore, the function is increasing for
b.
To find the interval on which the function
b.
Answer to Problem 5RE
The function is then decreasing for
Explanation of Solution
Given information:
The given function is
Formula:
Chain rule:
Consider the function
Using product and chain rule:
Here the second derivative is always negative, so the function is concave down for all values of
Here, the function is positive until one point which is at zero and then negative.
Therefore, the function is decreasing for
c.
To find the interval on which the function
c.
Answer to Problem 5RE
The function is never concave up.
Explanation of Solution
Given information:
The given function is
Formula:
Chain rule:
Consider the function
Using product and chain rule:
Here the second derivative is always negative, so the function is concave down for all values of
This has no solution so there are no values of
Therefore, the function is never concave up.
d.
To find the interval on which the function
d.
Answer to Problem 5RE
The function is always concave down.
Explanation of Solution
Given information:
The given function is
Formula:
Chain rule:
Consider the function
Using product and chain rule:
Here the second derivative is always negative, so the function is concave down for all values of
This has no solution so there are no values of
Therefore, the function is always concave down.
e.
To find the interval on which the function
e.
Answer to Problem 5RE
The function haslocal maximum at
Explanation of Solution
Given information:
The given function is
Formula:
Chain rule:
Consider the function
Using product and chain rule:
Equating the equation to zero,
Now to determine if the critical value
Since the derivative switches from positive to negative, the critical point is a local maximum. It is also an absolute maximum since there are no endpoints or other critical points.
Therefore, the function has local maximum at
f.
To find the interval on which the function
f.
Answer to Problem 5RE
The function has noinflection point.
Explanation of Solution
Given information:
The given function is
Formula:
Chain rule:
Consider the function
Using product and chain rule:
The point of inflection occurs at the values
Since the square root of a negative number is not a real number, there are no values of
Since there are no values of
Therefore, the function has no inflection point.
Chapter 5 Solutions
Calculus: Graphical, Numerical, Algebraic: Solutions Manual
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