To determine To sketch: The graph of the function f ( x ) = 2 x ( 1 − x ) by using graphing strategy. 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. Calculation: The given function is f ( x ) = 2 x ( 1 − x ) . Obtain the domain of the function as follows. 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, ( 1 − x ) = 0 x = 1 That is, the function is not defined when x = 1 . Therefore, the domain of the function is ℝ − { 1 } . Substitute x = 0 for the function to obtain the y -intercepts. f ( 0 ) = 2 ( 0 ) 1 − 0 = 0 Thus, the y -intercept of the function is ( 0 , 0 ) . Equate f ( x ) to 0 to obtain its x -intercepts. 2 x 1 − x = 0 2 x = 0 x = 0 Thus, the x -intercept is ( 0 , 0 ) . From the given function, it is noted that the numerator and the denominator have same degree. Thus, the horizontal asymptote can be obtained using a m x m b n x n . That is, 2 x − x = − 2 So the line y = − 2 is a horizontal asymptote. Since the function is undefined at the point x = 1 , it has a vertical asymptote at this point. Thus, the vertical asymptote of f ( x ) = 2 x ( 1 − x ) is x = 1 . The derivative of the function f ( x ) is, f ′ ( x ) = ( 1 − x ) 2 − 2 x ( − 1 ) ( 1 − x ) 2 = 2 − 2 x + 2 x ( 1 − x ) 2 = 2 ( 1 − x ) 2 Equate f ′ ( x ) to 0 to obtain the critical number. Observe that the function f ′ ( x ) is undefined at x = 1 . Thus, on equating the denominator of f ′ ( x ) to zero, the partition number is obtained as x = 1

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 1MP

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To sketch: The graph of the function f(x)=2x(1−x) by using graphing strategy.

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Chapter 11.3, Problem 70E

Chapter 11.4, Problem 2MP

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Chapter 11 Solutions

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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