Explanation Given: The function is y = x x 2 + 9 . Formula used: Test for concavity: Let f ″ ( x ) exist for all x ∈ ( a , b ) , then · y = f ″ ( x ) > 0 for all x ∈ ( a , b ) , here the function f is concave upward on ( a , b ) . · y = f ″ ( x ) < 0 for all x ∈ ( a , b ) , here the function f is concave downward on ( a , b ) . y = f ( x ) has a point of inflection if f ″ ( x ) = 0 . Calculation: The function is y = x x 2 + 9 . Let f ( x ) = x x 2 + 9 . … (1) First, find the derivative. f ′ ( x ) = ( x 2 + 9 ) d d x x − x d d x ( x 2 + 3 ) ( x 2 + 9 ) 2 = ( x 2 + 9 ) − x ( 2 x ) ( x 2 + 9 ) 2 = x 2 + 9 − 2 x 2 ( x 2 + 9 ) 2 = 9 − x 2 ( x 2 + 9 ) 2 Now, find the second derivative. f ″ ( x ) = ( x 2 + 9 ) 2 d d x ( 9 − x 2 ) − ( 9 − x 2 ) d d x ( x 2 + 9 ) 2 ( ( x 2 + 9 ) 2 ) 2 = ( x 2 + 9 ) 2 ( − 2 x ) − ( 9 − x 2 ) 2 ( x 2 + 9 ) 2 x ( ( x 2 + 9 ) 2 ) 2 = ( x 2 + 9 ) [ ( − 2 x ) ( x 2 + 9 ) − ( 9 − x 2 ) 4 x ] ( ( x 2 + 9 ) 2 ) 2 = 2 x [ − x 2 − 9 − 18 + 2 x 2 ] ( x 2 + 9 ) 3 = 2 x [ x 2 − 27 ] ( x 2 + 9 ) 3 Find the value of x by substitute f ″ ( x ) = 0 . 2 x [ x 2 − 27 ] ( x 2 + 9 ) 3 = 0 2 x [ x 2 − 27 ] = 0 That is, 2 x = 0 or x 2 − 27 = 0 x = 0 or x 2 = 27 x = 0 or x = ± 27 x = 0 or x = ± 3 3 Thus, the zeroes of the f ″ ( x ) are x = 0 , ± 3 3 . This divides the x − axis into four intervals: ( − ∞ , − 3 3 ) , ( − 3 3 , 0 ) , ( 0 , 3 3 ) and ( 3 3 , ∞ )

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Question

Chapter 4.4, Problem 16E

To determine

Write the intervals on which the function is concave up or down. Also, find the points of inflection.

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Chapter 4.4, Problem 15E

Chapter 4.4, Problem 17E

Chapter 4 Solutions

CALCULUS (CLOTH)

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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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY

Write the intervals on which the function is concave up or down. Also, find the points of inflection.

Solution Summary: The author explains how the function f is concave up and down, and the points of inflection.

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Write the intervals on which the function is concave up or down. Also, find the points of inflection.

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Chapter 4 Solutions