Explanation Given: The function y = x 4 − 3 x 3 + 4 x Key concepts used: Critical values of a function can be found by solving the equation f ' = 0 . The sign of f ' is positive when graph is increasing. The sign of f ' is negative when graph is decreasing. The sign of f ' ' is positive when graph is concave up. The sign of f ' ' is negative when graph is concave down. Maximum of a function occurs when derivative changes its sign from positive to negative. Minimum of a function occurs when derivative changes its sign from negative to positive. Inflection point occurs when double derivative changes its sign, provided the original function is defined at that point. Calculation: Let us first find critical values of the given function. For that, first we find the derivative of the function and set it equal to zero: ⇒ y = x 4 − 3 x 3 + 4 x ⇒ y ' = 4 x 3 − 9 x 2 + 4 ⇒ y ' = 4 x 3 − 9 x 2 + 4 = 0 ⇒ x = − 0.59 , 2 , 0.84 Thus, the critical values of the given function are x = − 0.59 , 2 , 0.84 Function is increasing when derivative is positive: ⇒ y ' > 0 ⇒ 4 x 3 − 9 x 2 + 4 > 0 ⇒ x ∈ ( − 0.59 , 0.84 ) ∪ ( 2 , ∞ ) Function is decreasing when derivative is negative: ⇒ y ' < 0 ⇒ 4 x 3 − 9 x 2 + 4 < 0 ⇒ x ∈ ( − ∞ , − 0

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 4.5, Problem 23E

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Chapter 4.5, Problem 22E

Chapter 4.5, Problem 24E

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Chapter 4 Solutions

Calculus - Standalone book

Ch. 4.1 - Prob. 1PQ Ch. 4.1 - Prob. 2PQ Ch. 4.1 - Prob. 3PQ Ch. 4.1 - Prob. 4PQ 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 Ch. 4.1 - Prob. 6E

Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E Ch. 4.1 - Prob. 11E Ch. 4.1 - Prob. 12E Ch. 4.1 - Prob. 13E Ch. 4.1 - Prob. 14E Ch. 4.1 - Prob. 15E Ch. 4.1 - Prob. 16E Ch. 4.1 - Prob. 17E Ch. 4.1 - Prob. 18E Ch. 4.1 - Prob. 19E Ch. 4.1 - Prob. 20E Ch. 4.1 - Prob. 21E Ch. 4.1 - Prob. 22E Ch. 4.1 - Prob. 23E Ch. 4.1 - Prob. 24E Ch. 4.1 - Prob. 25E Ch. 4.1 - Prob. 26E Ch. 4.1 - Prob. 27E Ch. 4.1 - Prob. 28E Ch. 4.1 - Prob. 29E Ch. 4.1 - Prob. 30E Ch. 4.1 - Prob. 31E Ch. 4.1 - Prob. 32E Ch. 4.1 - Prob. 33E Ch. 4.1 - Prob. 34E Ch. 4.1 - Prob. 35E Ch. 4.1 - Prob. 36E Ch. 4.1 - Prob. 37E Ch. 4.1 - Prob. 38E Ch. 4.1 - Prob. 39E Ch. 4.1 - Prob. 40E Ch. 4.1 - Prob. 41E Ch. 4.1 - Prob. 42E Ch. 4.1 - Prob. 43E Ch. 4.1 - Prob. 44E Ch. 4.1 - Prob. 45E Ch. 4.1 - Prob. 46E Ch. 4.1 - Prob. 47E Ch. 4.1 - Prob. 48E Ch. 4.1 - Prob. 49E Ch. 4.1 - Prob. 50E Ch. 4.1 - Prob. 51E Ch. 4.1 - Prob. 52E Ch. 4.1 - Prob. 53E Ch. 4.1 - Prob. 54E Ch. 4.1 - Prob. 55E Ch. 4.1 - Prob. 56E Ch. 4.1 - 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To graph: The given function.

Solution Summary: The author explains how critical values of a function can be found by solving the equation f'=0.

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To graph: The given function.

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To graph:

The given function.