Explanation Given: The function y = 4 − 2 x 2 + x 4 6 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 = 4 − 2 x 2 + x 4 6 ⇒ y ' = − 4 x + 2 x 3 3 ⇒ y ' = − 4 x + 2 x 3 3 = 0 ⇒ x = − 6 , 0 , 6 Function is increasing when derivative is positive: ⇒ y ' > 0 ⇒ − 4 x + 2 x 3 3 > 0 ⇒ x ∈ ( − 6 , 0 ) ∪ ( 6 , ∞ </

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 4.5, Problem 19E

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Chapter 4.5, Problem 18E

Chapter 4.5, Problem 20E

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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

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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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The given function.