Calculus: Early Transcendentals 9th Edition

Published by Cengage Learning
ISBN 10: 1337613924
ISBN 13: 978-1-33761-392-7

Chapter 4 - Section 4.1 - Maximum and Minimum Values - 4.1 Exercises - Page 287: 43

Answer

$x = 0,{\text{ }}x = 4,{\text{ }}x = \frac{4}{3}$

Work Step by Step

$$\eqalign{ & f\left( x \right) = {x^{1/3}}{\left( {4 - x} \right)^{2/3}} \cr & {\text{Differentiate with respect to }}x \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {{x^{1/3}}{{\left( {4 - x} \right)}^{2/3}}} \right] \cr & {\text{Using the product rule for derivatives}} \cr & f'\left( x \right) = {x^{1/3}}\frac{d}{{dx}}\left[ {{{\left( {4 - x} \right)}^{2/3}}} \right] + {\left( {4 - x} \right)^{2/3}}\frac{d}{{dx}}\left[ {{x^{1/3}}} \right] \cr & {\text{Computing derivatives and simplify}} \cr & f'\left( x \right) = {x^{1/3}}\left( {\frac{2}{3}} \right){\left( {4 - x} \right)^{ - 1/3}}\left( { - 1} \right) + {\left( {4 - x} \right)^{2/3}}\left( {\frac{1}{3}{x^{ - 2/3}}} \right) \cr & f'\left( x \right) = - \frac{2}{3}{x^{1/3}}{\left( {4 - x} \right)^{ - 1/3}} + \frac{1}{3}{x^{ - 2/3}}{\left( {4 - x} \right)^{2/3}} \cr & {\text{Factoring}} \cr & f'\left( x \right) = - \frac{1}{3}{x^{ - 2/3}}{\left( {4 - x} \right)^{ - 1/3}}\left( {2x - \left( {4 - x} \right)} \right) \cr & f'\left( x \right) = - \frac{1}{3}{x^{ - 2/3}}{\left( {4 - x} \right)^{ - 1/3}}\left( {2x - 4 + x} \right) \cr & f'\left( x \right) = - \frac{1}{3}{x^{ - 2/3}}{\left( {4 - x} \right)^{ - 1/3}}\left( {3x - 4} \right) \cr & f'\left( x \right) = - \frac{{3x - 4}}{{3{x^{2/3}}{{\left( {4 - x} \right)}^{1/3}}}} \cr & {\text{The derivative does not exist when the denominator is 0}} \cr & 3{x^{2/3}} = 0 \to x = 0 \cr & or \cr & {\left( {4 - x} \right)^{1/3}} = 0 \to x = 4 \cr & {\text{Find the values of }}x{\text{ where }}f'\left( x \right) = 0 \cr & - \frac{{3x - 4}}{{3{x^{2/3}}{{\left( {4 - x} \right)}^{1/3}}}} = 0 \cr & 3x - 4 = 0 \cr & x = \frac{4}{3} \cr & \cr & {\text{Applying the definition of critical numbers:}} \cr & {\text{A critical number of a function is a number }}c{\text{ in the domain }} \cr & {\text{of such that either }}f'\left( c \right) = 0{\text{ or }}f'\left( c \right){\text{ does not exist}}{\text{, so the}} \cr & {\text{critical numbers are:}} \cr & x = 0,{\text{ }}x = 4,{\text{ }}x = \frac{4}{3} \cr} $$
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