Concept explainers
Which of the following statements couldn’t possibly be true about the polynomial function P?
- (a) P has degree 3, two local
maxima , and twolocal minima . - (b) P has degree 3 and no local maxima or minima.
- (c) P has degree 4, one
local maximum , and no local minima.
(a)
To check: The given statement is true or not.
Answer to Problem 4E
The given statement is not true.
Explanation of Solution
Given:
The polynomial P has 3 degree. The given polynomial has two local maxima and two local minima.
Calculation:
Let us consider a polynomial P of degree n.
Then the graph of P has at most
Local extrema is the points on the graph that has maximum and minimum value.
If a polynomial has 3 degree then the no of extrema is 2.
The given statement is that polynomial has 2 local maxima and 2 local minima that means there are 4 local extrema, but the polynomial have only 2 extrema.
Thus, the given statement is not true.
(b)
To check: The given statement is true or not.
Answer to Problem 4E
The given statement is true.
Explanation of Solution
Given:
The polynomial P has 3 degree. The given polynomial has no local maxima or minima.
Calculation:
Let us consider a polynomial P of degree n.
Then the graph of P has at most
Local extrema is the points on the graph that has maximum and minimum value.
If a polynomial has 3 degree then at most 2 local extrema, but there is no minimum number for local minima and maxima.
So, it may be possible that there should not be any local maxima or minima.
Thus, the given statement is true.
(c)
To check: The given statement is true or not.
Answer to Problem 4E
The given statement is true.
Explanation of Solution
Given:
The polynomial P has 4 degree. The given polynomial has one local maximum and no minima.
Calculation:
Let us consider a polynomial P of degree n.
Then the graph of P has at most
Local extrema is the points on the graph that has maximum and minimum value.
If a polynomial has 4 degree then at most 3 local extrema, but there is no minimum number for local minima and maxima.
So, it may be possible that there should one local maximum and no local minima.
Thus, the given statement is true.
Chapter 3 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
- 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