Explanation Given: The function y = x + 16 − 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 + 16 − x ⇒ y ' = 1 2 x + − 1 2 16 − x ⇒ y ' = 1 2 x − 1 2 16 − x = 0 ⇒ x = 8 Thus, the critical value of the given function is x = 8 Function is increasing when derivative is positive: ⇒ y ' > 0 ⇒ 1 2 x − 1 2 16 − x > 0 ⇒ x ∈ ( 0 , 8 ) Function is decreasing when derivative is negative: ⇒ y ' < 0 ⇒ 1 2 x − 1 2 16 − x < 0 ⇒ x ∈ ( 8 , 16 ) In order to determine concavity, we check the sign of double derivative

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ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 4.5, Problem 28E

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Chapter 4.5, Problem 27E

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Chapter 4 Solutions

CALCULUS W/SAPLING ACCESS >IC<

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.

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To graph:

The given function.