Explanation Given information: The given curve is f ( x ) = 18 ( x − 3 ) ( x − 1 ) 2 3 . The given first and second order derivative of the curve are below: f ' ( x ) = 30 ( x − 9 5 ) ( x − 1 ) 1 3 .......... ( 1 ) f ' ' ( x ) = 20 ( x − 3 5 ) ( x − 1 ) 4 3 ......... ( 2 ) Calculation: In order to find the behavior of the curve, it is required to find the critical points, strictly increasing and strictly decreasing interval of the function, Maxima and Minima if exists. For critical points, substitute first order derivative to be equal to zero. Therefore from (1), f ' ( x ) = 0 30 ( x − 9 5 ) ( x − 1 ) 1 3 = 0 30 ( x − 9 5 ) = 0 x − 9 5 = 0 x = 9 5 x = 1.8 x = 1.8 is the critical point of thecurve. For x < 1 , consider the point x = 0 and check the value of first order derivative at that point. For the interval ( − ∞ , 1 ) ( f ' ( x ) ) x = 0 = 30 ( 0 − 9 5 ) ( 0 − 1 ) 1 3 ( f ' ( x ) ) x = 0 = 30 ( − 1.8 ) ( − 1 ) 1 3 ( f ' ( x ) ) x = 0 = − 5.4 ( − 1 ) ( f ' ( x ) ) x = 0 = 5.4 > 0 Since, the value of first order derivative at x = 0 is positive therefore, the behavior of curve is strictly increasing for the interval − ∞ < x < 1 . For 1 < x < 1.8 , consider the point x = 1.5 and check the value of first order derivative at that point. For the interval ( 1 , 1.8 ) ( f ' ( x ) ) x = 1.5 = 30 ( 1.5 − 9 5 ) ( 1.5 − 1 ) 1 3 ( f ' ( x ) ) x = 1.5 = 30 ( 1.5 − 1.8 ) ( 0.5 ) 1 3 ( f ' ( x ) ) x = 1

4th Edition

ISBN: 9781319308858

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 4.6, Problem 37E

To determine

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The behavior of the curve f(x)=18(x−3)(x−1)23.

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Chapter 4.6, Problem 36E

Chapter 4.6, Problem 38E

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If the price charged for a candy bar is p(x) cents, then x thousand candy bars will be sold in a certain city, where p(x)=158- X 10° a. Find an expression for the total revenue from the sale of x thousand candy bars. b. Find the value of x that leads to maximum revenue. c. Find the maximum revenue.

3 The total profit P(X) (in thousands of dollars) from the sale of x hundred thousand automobile tires is approximated by P(x) = -x³ + 12x² + 60x - 200, x≥5. Find the number of hundred thousands of tires that must be sold to maximize profit. Find the maximum profit. The maximum profit is $ when hundred thousand tires are sold.

Chapter 4 Solutions

CALCULUS:EARLY TRANS.(LL)-W/ACCESS

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