Explanation The polynomial function of degree n can have ( n − 1 ) zeros that is ( n − 1 ) turning points. The peak point or the highest point of the graph corresponds to relative maxima and the bottom point or the lowest point of the graph corresponds to relative minima . Use mathematical induction to prove the statement. Assume a polynomial function of degree n , for n ≥ 2 . P ( n ) = a n x n + a n − 1 x n − 1 + a n − 2 x n − 2 ⋯ + a 0 (1) Let n = 2 and solve for the number of extrema. Substitute 2 for n in above equation. P ( 2 ) = a 2 x 2 + a 1 x 1 + a 0 To find the number of extrema, differentiate the above equation with respect to x . P ′ ( 2 ) = d d x ( a 2 x 2 + a 1 x 1 + a 0 ) = d d x ( a 2 x 2 ) + d d x ( a 1 x 1 ) + d d x ( a 0 ) = 2 a 2 x + a 1 Equate the derivate to 0 to find the extrema. 2 a 2 x + a 1 = 0 x = − a 1 2 a 2 For n = 2 , the number of extrema is 1. So, the statement is true for initial condition, n = 2 . Not let the statement be correct for n = k . Substitute k for n in equation (1). P ( k ) = a k x k + a k − 1 x k − 1 + a k − 2 x k − 2 + ⋯ + a 0 (2) To find the number of extrema, differentiate the above equation with respect to x . P ′ ( k ) = d d x ( a k x k + a k − 1 x k − 1 + a k − 2 x k − 2 + ⋯ + a 0 ) = a k k x k − 1 + ( k − 1 ) a k − 1 x k − 2 + ⋯ + a 1 Number of extrema, for P ( k ) is k − 1

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 4.1, Problem 103E

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Chapter 4.1, Problem 102E

Chapter 4.1, Problem 104E

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Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

Ch. 4.1 - Explain each of the following: a. f is increasing...Ch. 4.1 - Prob. 2CQ Ch. 4.1 - Prob. 3CQ Ch. 4.1 - Prob. 4CQ Ch. 4.1 - Prob. 5CQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E

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Whether the statement “A polynomial function of degree n ( n > 2 ) can have at most ( n − 1 ) relative extrema” is true or false.

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Whether the statement “A polynomial function of degree n(n>2) can have at most (n−1) relative extrema” is true or false.

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Whether the statement “A polynomial function of degree n ( n &gt; 2 ) can have at most ( n − 1 ) relative extrema” is true or false.

Solution Summary: The author explains whether the statement "A polynomial function of degree n(n>2) can have at most

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Whether the statement “A polynomial function of degree n(n>2) can have at most (n−1) relative extrema” is true or false.

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Chapter 4 Solutions

Whether the statement “A polynomial function of degree n ( n > 2 ) can have at most ( n − 1 ) relative extrema” is true or false.