Explanation The given function is h ( x ) = ( x − 1 ) 2 / 3 + 1 (1) The domain of the function is the set of all real numbers except the number that is undefined on the function. As it is defined for all numbers on the given function, the domain of the given function is ( − ∞ , ∞ ) . It is known that y = h ( x ) . Substitute 0 for x in the given function to calculate the y -intercept. y = ( 0 − 1 ) 2 / 3 + 1 = ( − 1 ) 2 / 3 + 1 = 1 + 1 = 2 Substitute 0 for y in the function to calculate the x -intercept. 0 = ( x − 1 ) 2 / 3 + 1 ( x − 1 ) 2 / 3 = − 1 x − 1 = − 1 x = 0 This shows that the x -intercept is 0 and y -intercept is 2 . To determine the behavior of the function for large absolute values of x , take the limit of x to ∞ and − ∞ . Suppose that x approaches ∞ . lim x → ∞ h ( x ) = lim x → ∞ [ ( x − 1 ) 2 / 3 + 1 ] = ∞ Suppose that x approaches − ∞ . lim x → − ∞ h ( x ) = lim x → − ∞ [ ( x − 1 ) 2 / 3 + 1 ] = ∞ Thus, the graph of h approaches to ∞ as x becomes arbitrarily large. Since, the function has no line for x at ∞ , there is no horizontal asymptote. Since, the function tends to ∞ for no value of x , there is no vertical asymptote. Differentiate equation (1) with respect to x . h ′ ( x ) = ( 2 3 ) ( x − 1 ) 2 3 − 1 d d x ( x − 1 ) = ( 2 3 ) ( x − 1 ) 2 − 3 3 ( 1 ) = 2 3 ( x − 1 ) − 1 3 (2) This shows that for h ′ ( x ) = 0 , and the critical point comes out to be, 2 3 ( x − 1 ) − 1 3 = 0 x − 1 = 0 x = 1 Thus, the intervals are ( − ∞ , 1 ) and ( 1 , ∞ ) . Take the test points for different intervals in the table below to determine the signs of h ′ ( x ) . Intervals Test point c h ′ ( c ) Sign of h ′ ( x ) ( − ∞ , 1 ) − 3 − 0

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

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ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 4.3, Problem 59E

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Chapter 4.3, Problem 58E

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Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

Ch. 4.1 - Explain each of the following: a. f is increasing...Ch. 4.1 - Prob. 2CQ Ch. 4.1 - Prob. 3CQ Ch. 4.1 - Prob. 4CQ Ch. 4.1 - Prob. 5CQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E

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The graph of the function h ( x ) = ( x − 1 ) 2 / 3 + 1 by curve-sketching guide.

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