Concept explainers
To explain: the local extremums for a polynomial function of degree 3, 4, 5, and 6. Also describe the end behavior of the function.
Explanation of Solution
A polynomial function is of the form
Terms of a polynomial function should be arranged in descending order according to its degree to express it in a standard form and degree of each term should be a positive integer or whole number. The coefficients should be real numbers.
Leading coefficient of a polynomial function is the coefficient of the leading term.
Degree of the polynomial is the degree of leading term or the height degree in the polynomial function.
For the polynomial
The end behavior can describe the graph of a polynomial function as
The end behavior of a polynomial function can be determined by the leading coefficient and the degree of the polynomial.
If degree is even and leading coefficient is negative.
If degree is odd and leading coefficient is negative.
If degree is odd and leading coefficient is positive.
If degree is even and leading coefficient is positive.
Consider the graph of a polynomial function degree six.
An n degree polynomial can at most have
So, a 3 degree polynomial can have at most have
A 5 degree polynomial can have at most have
End behavior of an even degree and odd degree polynomial are similar.
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