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 ) = x 3 ln x . The domain of the function f ( x ) = x 3 ln x is obtained as follows, The function ln is defined only on the interval ( 0 , ∞ ) . So, the function f ( x ) = x 3 ln x is also defined on the interval ( 0 , ∞ ) . Thus, the domain of the function is ( 0 , ∞ ) . Equate f ( x ) to 0 to obtain its x -intercepts. x 3 ln ( x ) = 0 x = 1 Thus, the x -intercept of the function is ( 1 , 0 ) The function is not defined at the point x = 0 . So, the function has no y -intercept. Since lim x → ∞ ( x 3 ln x ) = ∞ , it has no horizontal asymptote. Also for every finite value, the function also attains finite value. So it has no vertical asymptote. The derivative of the function f ( x ) is, f ′ ( x ) = d d x ( x 3 ln x ) = 3 x 2 ln ( x ) + x 3 x = 3 x 2 ln ( x ) + x 2 Equate f ′ ( x ) to 0 to obtain the critical number. x 2 ( 3 ln x + 1 ) = 0 3 ln x = − 1 ln x = − 1 3 x = e − 1 3 The partition points of f ′ ( x ) is x = e − 1 3 . The sign chart of f ′ ( x ) is as shown below in Table 1. Interval Test number f ′ ( x ) 0 < x < e − 1 3 0.2 f ′ ( 0.2 ) = ( 0.2 ) 2 ( 3 ln ( 0.2 ) + 1 ) = − 0.153 f ′ ( x ) < 0 decreasing e − 1 3 < x < ∞ 1 f ′ ( 1 ) = ( 1 ) 2 ( 3 ln ( 1 ) + 1 ) = 1 f ( x ) > 0 increasing Table 1 From Table 1, it is observed that the function is decreasing on ( 0 , e − 1 3 ) and increasing on ( e − 1 3 , ∞ ) . The function has local minimum at x = e − 1 3 . Substitute x = e − 1 3 in f ( x ) and obtain its minimum value. f ( e − 1 3 ) = ( e − 1 3 ) 3 ln ( e − 1 3 ) = − 1 3 e Thus, the local minimum is ( e − 1 3 , − 1 3 e )

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

14th Edition

ISBN: 9780134668574

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

Publisher: PEARSON

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Chapter 4, Problem 30RE

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Chapter 4 Solutions

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

Ch. 4.1 - Matched Problem 1 Repeat Example 1 for f(x) = x2 ...Ch. 4.1 - Matches Problem 2 Find the critical numbers of f,...Ch. 4.1 - Matched Problem 3 Find the critical numbers of f,...Ch. 4.1 - Matched Problem 4 Find the critical numbers of f,...Ch. 4.1 - Matched Problem 5 Find the critical numbers of f,...Ch. 4.1 - Prob. 6MP Ch. 4.1 - Matched Problem 7 Given f(x) = x3 9x2 + 24x 10,...Ch. 4.1 - Matched Problem 8 The graph in Figure 13...Ch. 4.1 - Matched Problem 9 The graph of the total revenue...Ch. 4.1 - Skills Warm-up Exercises In Problems 1-8, inspect...

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

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