Explanation Given: The function is y = x ( x − 8 x ) , x ≥ 0 . 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 − 8 x ) , x ≥ 0 . We can rewrite it as: y = x 2 − 8 x x = x 2 − 8 x 3 2 Let f ( x ) = x 2 − 8 x 3 2 . … (1) First, find the derivative. f ′ ( x ) = 2 x 2 − 1 − 8 ( 3 2 ) x 3 2 − 1 = 2 x − 12 x 1 2 Now, find the second derivative. f ″ ( x ) = 2 − 12 ( 1 2 ) x 1 2 − 1 = 2 − 6 x 1 2 Find the value of x by substitute f ″ ( x ) = 0 . 2 − 6 x 1 2 = 0 − 6 x 1 2 = − 2 6 2 = x 1 2 3 = x 1 2 Take square on both sides

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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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 11E

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 10E

Chapter 4.4, Problem 12E

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

Solution Summary: The author explains how to determine whether the function is concave up or down, and find 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