Explanation Given: The function f ( x ) = x 3 − 3 x 2 + 6 x Key concepts used: In order to decide the sign of the first derivative, we look where the graph is increasing and where it is decreasing. The sign of f ' is positive when graph is increasing. The sign of f ' is negative when graph is decreasing. In order to decide the sign of the double derivative, we look where the graph is concave up and where it is concave down. 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. Calculation: Let us first find intervals where the function is increase and where it is decreasing. For that, we will first find the derivative: f ( x ) = x 3 − 3 x 2 + 6 x ⇒ f ' ( x ) = 3 x 2 − 6 x + 6 Function is increasing when derivative is positive: ⇒ f ' ( x ) > 0 ⇒ 3 x 2 − 6 x + 6 > 0 ⇒ 3 ( x 2 − 2 x + 2 ) > 0 ⇒ x ∈ ( <

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 4.5, Problem 12E

To determine

To graph:

The given function.

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Chapter 4.5, Problem 11E

Chapter 4.5, Problem 13E

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Chapter 4 Solutions

CALCULUS (CLOTH)

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To graph: The given function.

Solution Summary: The author explains that the given function doesn't have any local extremes as the function is always increasing and the derivative never changes its sign.