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ber of critical points. The function f (x) = x has no extrema on any open interval. Every nth-degree polynomial h00 (

ber of critical points. The function f (x) = x has no extrema on any open interval. Every nth-degree polynomial h00 (

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Please solve 108, 110, and 112

### Creating Polynomial Functions In Exercises 103–106, find a polynomial function \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_2 x^2 + a_1 x + a_0 \) that has only the specified extrema. **(a)** Determine the minimum degree of the function and give the criteria you used in determining the degree. **(b)** Using the fact that the coordinates of the extrema are solution points of the function, and that the x-coordinates are critical numbers, determine a system of linear equations whose solution yields the coefficients of the required function. **(c)** Use a graphing utility to solve the system of equations and determine the function. **(d)** Use a graphing utility to confirm your result graphically. - **103.** Relative minimum: (0, 0); Relative maximum: (2, 2) - **104.** Relative minimum: (0, 0); Relative maximum: (4, 1000) - **105.** Relative minima: (0, 0), (4, 0); Relative maximum: (2, 4) - **106.** Relative minimum: (1, 2); Relative maxima: (–1, 4), (3, 4) ### True or False? In Exercises 107–112, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. - **107.** There is no function with an infinite number of critical points. - **108.** The function \( f(x) = x \) has no extrema on any open interval. - **109.** Every nth-degree polynomial has \( (n - 1) \) critical numbers. - **110.** An nth-degree polynomial has at most \( (n - 1) \) critical numbers. - **111.** There is a relative extremum at each critical number. - **112.** The relative maxima of the function \( f \) are \( f(1) = 4 \) and \( f(3) = 10 \). Therefore, \( f \) has at least one minimum for some \( x \) in the interval \((1, 3)\). ### Graphs and Diagrams The page also includes diagrams and graphs illustrating
Transcribed Image Text:### Creating Polynomial Functions In Exercises 103–106, find a polynomial function \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_2 x^2 + a_1 x + a_0 \) that has only the specified extrema. **(a)** Determine the minimum degree of the function and give the criteria you used in determining the degree. **(b)** Using the fact that the coordinates of the extrema are solution points of the function, and that the x-coordinates are critical numbers, determine a system of linear equations whose solution yields the coefficients of the required function. **(c)** Use a graphing utility to solve the system of equations and determine the function. **(d)** Use a graphing utility to confirm your result graphically. - **103.** Relative minimum: (0, 0); Relative maximum: (2, 2) - **104.** Relative minimum: (0, 0); Relative maximum: (4, 1000) - **105.** Relative minima: (0, 0), (4, 0); Relative maximum: (2, 4) - **106.** Relative minimum: (1, 2); Relative maxima: (–1, 4), (3, 4) ### True or False? In Exercises 107–112, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. - **107.** There is no function with an infinite number of critical points. - **108.** The function \( f(x) = x \) has no extrema on any open interval. - **109.** Every nth-degree polynomial has \( (n - 1) \) critical numbers. - **110.** An nth-degree polynomial has at most \( (n - 1) \) critical numbers. - **111.** There is a relative extremum at each critical number. - **112.** The relative maxima of the function \( f \) are \( f(1) = 4 \) and \( f(3) = 10 \). Therefore, \( f \) has at least one minimum for some \( x \) in the interval \((1, 3)\). ### Graphs and Diagrams The page also includes diagrams and graphs illustrating
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