Explanation Definition used: If f ′ ( x ) > 0 on an interval ( a , b ) , then the graph of f ( x ) increases on that interval and if f ′ ( x ) < 0 , then f ( x ) decreases on that interval. Concavity: For the interval ( a , b ) , if f ″ ( x ) > 0 , then the function f is concave upward and if f ″ ( x ) < 0 , then f is concave downward. Oblique asymptote: If a graph approaches a line that is neither horizontal nor vertical as x approaches ∞ and − ∞ , then that line is called an oblique asymptote. Result used: The function f ( x ) is a rational function for which the degree of numerator is 1 more than the degree of denominator. Then, using long division method f ( x ) can be written as f ( x ) = m x + b + r ( x ) d ( x ) , where degree of r ( x ) is less than degree of d ( x ) . The line y = m x + b is an oblique asymptote for the graph of f ( x ) . Calculation: Domain: The given function is f ( x ) = x 3 3 − x 2 . Since, the given f ( x ) is a rational function, its domain is the set of all real numbers other than the roots of its denominator. Equate the denominator to 0 in order to obtain its undefined points, 3 − x 2 = 0 x 2 = 3 x = ± 3 The denominator becomes zero when x = ± 3 . So, the function is not defined at − 3 and 3 . Thus, the domain of the function is all real numbers except ± 3 . y- intercepts: Set x = 0 and find the y −intercepts. f ( 0 ) = 0 3 3 − 0 2 = 0 3 f ( 0 ) = 0 Thus, the y -intercept of the function is ( 0 , 0 ) . x- intercepts: substitute f ( x ) = 0 in the given function, x 3 3 − x 2 = 0 x 3 = 0 x = 0 Thus, the x -intercept of the function is ( 0 , 0 ) . Asymptotes: To prove that y = x is an oblique asymptote. Apply polynomial long division, it gives the result as f ( x ) = x 3 3 − x 2 . That is, f ( x ) = − x + 3 x − x 2 + 3 . Thus, by using the above result, as x → ± ∞ , y = f ( x ) → − x . Therefore, y = − x is an oblique asymptote of the function f ( x ) = x 3 3 − x 2 . The function is not defined at x = ± 3 . So, as x tends to − 3 or 3 , the value of the function will tends to infinity. Thus, the vertical asymptotes are x = − 3 and x = 3 First derivative test: The derivative of the function f ( x ) is, f ′ ( x ) = d d x ( x 3 3 − x 2 ) = 9 x 2 − x 4 ( 3 − x 2 ) 2 Set f ′ ( x ) = 0 and find the critical points. 9 x 2 − x 4 = 0 x 2 ( 9 − x 2 ) = 0 x = 0 , ± 3 Thus, the critical points of f is x = 0 , x = 3 and x = − 3 . Since the function is not defined at x = ± 3 , the partition numbers are x = 0 , x = − 3 , x = − 3 , x = 3 and x = 3 . The sign chart of f ′ ( x ) is as shown below in Table 1. Interval Test numbers f ′ ( x ) − ∞ < x < − 3 −4 f ′ ( − 4 ) = 9 ( − 4 ) 2 − ( − 4 ) 4 ( 3 − ( − 4 ) 2 ) 2 = − 0.66 f ′ ( x ) < 0 decreasing − 3 < x < − 3 −2 f ′ ( − 2 ) = 9 ( − 2 ) 2 − ( − 2 ) 4 ( 3 − ( − 2 ) 2 ) 2 = 20 f ′ ( x ) > 0 increasing − 3 < x < 0 −0.5 f ′ ( − 0.5 ) = 9 ( − 0.5 ) 2 − ( − 0.5 ) 4 ( 3 − ( − 0.5 ) 2 ) 2 = 0.289 f ′ ( x ) > 0 increasing 0 < x < 3 0.5 f ′ ( 0.5 ) = 9 ( 0.5 ) 2 − ( 0.5 ) 4 ( 3 − ( 0.5 ) 2 ) 2 = 0.289 f ′ ( x ) > 0 increasing 3 < x < 3 2 f ′ ( 2 ) = 9 ( 2 ) 2 − ( 2 ) 4 ( 3 − ( 2 ) 2 ) 2 = 20 f ′ ( x ) > 0 increasing 3 < x < ∞ 4 f ′ ( 4 ) = 9 ( 4 ) 2 − ( 4 ) 4 ( 3 − ( 4 ) 2 ) 2 = − 0

College Mathematics for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134674148

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Chapter 11.4, Problem 61E

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To sketch: The graph of the function f(x)=x33−x2 by using graphing strategy.

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Chapter 11.4, Problem 60E

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College Mathematics for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

Ch. 11.1 - Matched Problem 1 Repeat Example 1 for f(x) = x2 –...Ch. 11.1 - Matches Problem 2 Find the critical numbers of f,...Ch. 11.1 - Matched Problem 3 Find the critical numbers of f,...Ch. 11.1 - Prob. 4MP Ch. 11.1 - Prob. 5MP Ch. 11.1 - Prob. 6MP Ch. 11.1 - Prob. 7MP Ch. 11.1 - Prob. 8MP Ch. 11.1 - Prob. 9MP Ch. 11.1 - Skills Warm-up Exercises In Problems 1-8, inspect...

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The graph of the function f ( x ) = x 3 3 − x 2 by using graphing strategy.

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To sketch: The graph of the function f(x)=x33−x2 by using graphing strategy.

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Chapter 11 Solutions