Explanation Given information: The curve function is f ( x ) = x 5 − 5 x 4 − x 3 + 28 x 2 − 2 x (1) Calculation: Equate Equation (1) equal to 0. f ( x ) = 0 x 5 − 5 x 4 − x 3 + 28 x 2 − 2 x = 0 (2) Solve Equation (2). x = 0 , − 2.09 , 0.07 Draw the graph of the function f by substituting different values for x . The sketches of the function f with the important aspects are shown in Figure 1, Figure 2, and Figure 3. Differentiate Equation (1) with respect to x . f ' ( x ) = 5 x 4 − 20 x 3 − 3 x 2 + 56 x − 2 (3) Equate Equation (3) equal to 0. f ' ( x ) = 0 5 x 4 − 20 x 3 − 3 x 2 + 56 x − 2 = 0 (4) Solve Equation (4). x = − 1.50 , 0.04 , 2.62 , 2.84 The sketches of the function f ' with the important aspects are shown in Figure 4 and Figure 5. Differentiate Equation (3) with respect to x . f " ( x ) = 20 x 3 − 60 x 2 − 6 x + 56 (5) Equate Equation (5) equal to 0. f " ( x ) = 0 20 x 3 − 60 x 2 − 6 x + 56 = 0 (6) Solve Equation (6). x = − 0.89 , 1.15 , 2.74 The sketch of the function f " is shown in Figure 6. Refer Figure 4 and Figure 5. The curve f is decreasing on ( − 1.50 , 0.04 ) and ( 2.62 , 2.84 ) . The curve f is increasing on ( − ∞ , − 1.50 ) , ( 0.04 , 2.62 ) , and ( 2.84 , ∞ ) . The local minimum value occurs at x = 0.04 and x = 2.84 . The local maximum value occurs at x = − 1.50 and x = 2.62 . Therefore, the intervals of increase is on ( − ∞ , − 1.50 ) _ , ( 0.04 , 2.62 ) _ , and ( 2.84 , ∞ ) _ . Therefore, the intervals of decrease is on ( − 1.50 , 0.04 ) _ and ( 2.62 , 2.84 ) _ . Find the extreme values (local maxima and local minima ) as shown below: Substitute 0.04 for x in Equation (1). f ( 0.04 ) = ( 0.04 ) 5 − 5 ( 0.04 ) 4 − ( 0.04 ) 3 + 28 ( 0.04 ) 2 − 2 ( 0.04 ) = − 0.04 Substitute 2.84 for x in Equation (1). f ( 2.84 ) = ( 2.84 ) 5 − 5 ( 2

Single Variable Calculus: Early Transcendentals, Volume I

8th Edition

ISBN: 9781305270343

Author: James Stewart

Publisher: Cengage Learning

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Chapter 4.6, Problem 1E

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To sketch: The graph of f that reveals all important aspects of the curve and the graph of f' and f" to estimate the intervals of increase, decrease, the extreme values, the intervals of concavity and the inflection points.

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Chapter 4.5, Problem 76E

Chapter 4.6, Problem 2E

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Chapter 4 Solutions

Single Variable Calculus: Early Transcendentals, Volume I

Ch. 4.1 - Explain the difference between an absolute minimum...Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Use the graph to state the absolute and local...Ch. 4.1 - Use the graph to state the absolute and local...Ch. 4.1 - Sketch the graph of a function f that is...Ch. 4.1 - Sketch the graph of a function f that is...Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E

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The graph of f that reveals all important aspects of the curve and the graph of f ' and f " to estimate the intervals of increase, decrease, the extreme values, the intervals of concavity and the inflection points.

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To sketch: The graph of f that reveals all important aspects of the curve and the graph of f' and f" to estimate the intervals of increase, decrease, the extreme values, the intervals of concavity and the inflection points.

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Chapter 4 Solutions