a.
Find the intervals on which the function is increasing using analytical method.
a.
Answer to Problem 14RE
Function
Explanation of Solution
Given:
Function
Concept used:
If the derivative of a function is positive on an interval, then the function is increasing on that interval.
Also, if the derivative of a function is negative on an interval, then the function is decreasing on that interval.
Calculation:
First derivative of
For solution put
Thus,
According to a known result
Thus,
So,
Conclusion:
Function
b.
Find the intervals on which the function is decreasing using analytical method.
b.
Answer to Problem 14RE
Function
Explanation of Solution
Given:
Function
From part (a), first derivative of
Concept used:
If the derivative of a function is positive on an interval, then the function is increasing on that interval.
Also, if the derivative of a function is negative on an interval, then the function is decreasing on that interval.
Calculation:
First derivative of
According to a known result
Thus,
So,
Conclusion:
Function
c.
Find the intervals on which the function is concave up using analytical method.
c.
Answer to Problem 14RE
Function
Explanation of Solution
Given:
Function
First derivative of
Concept used:
If the second derivative of a function is positive on an interval, then the function is concave up on that interval.
Also, if the second derivative of a function is negative on an interval, then the function is concave down on that interval.
Calculation:
First derivative of
Second derivative of
Differentiate the
For solution put
Since,
According to a known result
Thus,
So,
Conclusion:
Function
d.
To Find: the intervals on which the function is concave down using analytical method.
d.
Answer to Problem 14RE
Function
Explanation of Solution
Given:
Function
First derivative of
Second derivative of
Concept used:
If the second derivative of a function is positive on an interval, then the function is concave up on that interval.
Also, if the second derivative of a function is negative on an interval then the function is concave down on that interval.
Calculation:
Second derivative of
For solution put
Since,
According to a known result
Thus,
So,
Graph of function
:
Conclusion:
Function
e.
Find local extreme values using the graph.
e.
Answer to Problem 14RE
Function has
Function has
Explanation of Solution
Given:
Function
Graph of function
Calculation:
According to the graph,
Function has local
Function has local
Conclusion:
Function has local maxima at
Function has local minima at
f.
Find inflection points using the graph.
f.
Answer to Problem 14RE
Inflection point
Explanation of Solution
Given:
Function
Second derivative of
Concept used:
Inflection point: A point of inflection on a curve is a continuous point at which the function changes its concavity.
Calculation:
Second derivative of
For inflection point put
Inflection point:
Conclusion:
Inflection point is
Chapter 4 Solutions
Advanced Placement Calculus Graphical Numerical Algebraic Sixth Edition High School Binding Copyright 2020
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning