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